Aqt instructor-notes-final

ADVANCED QUANTITATIVE TECHNIQUES LECTURES & NOTES

Lectures & Notes

ADVANCED QUANTITATIVE TECHNIQUES(COURSE FOR PHD STUDENTS)

ByDr. Anwar F. Chishti

ProfessorFaculty of Management & Social

Sciences

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ADVANCED QUANTITATIVE TECHNIQUES

Course PlanFall Semester 2012

Course Instructor Professor Dr. Anwar F. Chishti

Contacts: Phone Phone: 0346-9096046

Email [email protected]; [email protected]

Class venue Computer Laboratory

Course contentsTopic 1: Simple/Two-Variable Regression Analysis:

An introduction of estimated model and its interpretation, Regression Coefficients and Related Diagnostic Statistics:

Computational Formulas Evaluating the results of regression analysis Standard assumptions, BLUE properties of the estimator. Take-home assignment - 1

Topic 2: Simple Regression to Multiple Regression Analysis

Shortcomings of simple/two-variables regression analysis An example of multiple regression analysis Use of Likert-scale type questionnaire, raw-data entry, reliability test

and generation of variables Estimation of multiple regression model Evaluation of the estimated model in terms of F-statistic, R2 and t-

statistic/p-value Take-home assignment - 2

Topic 3: Multiple Regression: Model specification

3.1(a) Conceiving research ideas and converting it into research projects: a procedure

3.1(b) Incorporating theory as the base of your research: econometrics theory & economics/management theory

Take-home assignment – 3(a) 3.2 (a) Specification of an econometric model: mathematical

specification 3.2(b) Some practical examples of mathematical specification:

production-function specification, cost-function specification, revenue-function specification

Take-home assignment – 3(b)

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mailto:[email protected]



3.3(a) Conceptual/econometric modeling: (a) Examples in Finance; (b) Examples in Marketing; (c) Examples in HRM

3.3(b) Incorporating theory as the base of your research: econometrics theory & economics/management theory

Take-home assignment: adopting, adapting and developing a new questionnaire

Topic 4: Analyzing mean values

Analyzing mean value, using one-sample t-test Comparing mean-differences of two or more groups Comparing two groups

* Independent samples t test* Paired-sample t test

Comparing more-than-two groups* One-Way ANOVA* Repeated ANOVA

Take-home assignment – 4

Topic 5: Uses of estimated econometric models

Some examples Take-home assignment – 5

Topic 6: Relaxing of Standard Assumptions: Normality Assumption and its testing

Normality assumption Its testing Take-home assignment – 6

Topic 7: Problem of Multicollinearity: What Happens if Regressors are Correlated?

Consequences, tests for detection and solutions/remedies Take-home assignment - 7

Topic 8: Problem of Heteroscadasticity: What Happens if the Error Variance is

nonconstant? Consequences, tests for detection and solutions/remedies Take-home assignment - 8

Topic 9: Problem of Autocorrelation: What Happens if the Error terms are correlated?

Consequences, tests for detection and solutions/remedies Take-home assignment - 9

Topic 10: Mediation and moderation analysis - I

Estimating and testing mediation Take-home assignment – 10

Topic 11: Mediation and moderation analysis - II

Estimating and testing moderation Take-home assignment – 9

Topic 12: Time-series analysis - I

Unit root analysis

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Take-home assignment – 10

Topic 13: Time-series analysis - II

Unit root, co-integration and error correction modeling (ECM) Take-home assignment – 11

Topic 14 Panel data analysis, Simultaneous equation models/Structural equation models

Panl data analysis SEM, ILS, 2SLS and 3SLS Take-home assignment – 12

Topic 15 Qualitative response regression models (when dependent variables are binary/dummy) and Optimization

LPM, Logit model and Probit Model Take-home assignment – 13(a) * Optimization: minimization and maximization Take-home assignment – 13(b)

Topic 16 Welfare analysis: maximization of producer and consumer surpluses and minimization of social costs

Required Text & Recommended Reading

The prescribed textbooks for this course are:

Gujarati, Damodar N. Basic Econometrics, 4th Edition. McGraw-Hill. 2007

Stock, J. H. and Watson, M.W. Introduction to Econometrics, 3/E. Pearson Education, 2011

Reference Books/Materials

Studenmund, A.H. Using Econometrics: A Practical Guide, 6/E, Prentice Hall

Asteriou, D. and Hall, S.G. Applied Econometrics – A Modern Approach. Palgrave Macmillan, 2007.

Andren, Thomas. (2007). Econometrics. Bookboon.com

Salvatore, D and Reagle, D. Statistics and Econometrics, 2nd Ed. Schaum’s Outlines.

Instructor’s class-notes (hard copy at photo-copier shop)

Assessment Criteria

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Details Due Date Weighting

Individual Assignments10 best weekly assignments (out of total 13 - 15, each having 2 marks) will be counted toward total 20% marks.

20 %

Group research on selected research topics

A group of 2 students will select a topic, carry out research, complete a research study, and make presentation in during the last classes of the semester

20 %

Mid-term Examination As per University’s announcement 20 %

Final Examination As per University’s announcement 40 %

Total marks: 100

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Topic 1Simple/Two-Variable Regression Analysis

1.1 Simple regression analysis: an example

Assuming a survey of 10 families yields the following data on their consumption expenditure (Y)

and income (X).

Y (Thousands) X (Thousands) 70 80 65 100 90 120 95 140 110 160 115 180 120 200 140 220 155 240 150 260The theory suggests that families’ consumption (Y) depends on their income (X); hence,

econometric model may be specified, as follows.

Y = f(X) (General form) (1a)Or Y = β0 + β1X + e (Linear form) (1b)

The above stated regression analysis model contains two variables (one independent variable X

and one dependent variable Y); this model is therefore called Two-variables or Simple regression

analysis model.

Is this type of Simple or Two-variable model justified? We will discuss this question later on;

let’s first estimate this model, using the Statistical Package for Social Sciences’ software SPSS.

The estimated model & interpretation

Y = 24.4530 + 0.5091 X (2a) (6.4140) (0.0357) (Standard Error) (2b) (3.8124) (14.2445) (t-statistic) (2c)

(0.005) (0.000) (p-value/sig. level) (2d) R= 0.981 R2 = 0.9621 R2

adjusted = 0.957 F = 203.082 (p-value = 0.000) DW = 2.6809 N = 10 (2e)

1.2 Regression analysis: computational formulas

The econometric model specified in (1) is estimated in the form of estimated model (2a) along

with all its diagnostic statistics 2(b – e), using the formulas provided, as follows.

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The coefficients ßs

(3)

(4)

(5)

Variances (σ 2) and Standard Errors (S.E):

(6)

(7)

(8)

(9)

(10)

T-ratios:

(11)

(12)

The Coefficient of Determination ( R2 ):

(13)

F – Statistics:

(14) Durban-Watson (D.W) Statistics:

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(15)

1.3 Estimation of the model using computational formulas

We now use formula provided in (3) to (15), make computations like Table 3.3 (Gujarati,

2007) and resolve the model, as follows.

Yi = ßo + ß1 Xi + ℮i …….. Linear model (16)

Regression Coefficients ( ß i ):

(17)

(18)

Variances (σ 2) and Standard Errors (S.E):

(19)

(20)

(21)

(22)

(23)

T-ratios:

(24)

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(25) The Coefficient of Determination ( R2 ):

(26) F – Statistics:

(27)

The estimated model: Y = 24.4530 + 0.5091X (6.414) (0.0357) S.E.

(3.812) (14.244) t-ratio (0.005) (0.0000) (p-valuel)

R2 = 0.9621 F = 203.082 N = 10 (28)

1.4 Regression analysis: the underlying theory

The above reported formulas reflect how various needed computations are carried out in

regression analysis. Specifically, formula (4) estimates the coefficient (β1) of explanatory

variable X:

That is: ‘the deviations of individual observation on Xi from its mean, multiplied by deviations of

respective Yi from its mean (cross-deviation), divided by the squares of the variations of Xi’; so

it is the ratio between cross-deviations of X – Y variables and X variable. Theoretically, β1

measures ‘total cross deviations/variations per unit of variation in X-variable’. The intercept β0

measures ‘mean value of Y minus total contribution of mean of X’.

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1.5 Error term: its estimation and importance

When an econometric model, like 1(b), is specified:

Y = β0 + β1X + e (29a)

It contains an error or residual term (e); but when model is estimated like 2(a):

Y = 24.4530 + 0.5091X (29b)

The error term (e) seems to disappear; where does the error term go?

In fact the estimated model like 29(b) is valid only for the mean/average values of X and Y, and

equality in 29(b) does not hold when values other-than-mean values are used; we can compute

values of error terms or residuals, using the following formula.

Yi – Y = e (30a)

Yi – (24.4530 + 0.5091Xi) = e (30b)

Putting individual-observation values from the original data, that is:

Y X 70 80 65 100 90 120 95 140 110 160 115 180 120 200 140 220 155 240 150 260

Yi – (24.4530 + 0.5091Xi) = e

70 – (24.4530 + 0.5091*80 = 4.8181 (30c)

65 – (24.4530 + 0.5091*100) = -10.3636 (30d)

90 – (24.4530 + 0.5091*120 = 4.4545 (30e)

95 – (24.4530 + 0.5091*140) = -0.7272 (30f)

110 – (24.4530 + 0.5091*160) = 4.0909 (30g)

115 – (24.4530 + 0.5091*180) = -1.0909 (30i)

120 – (24.4530 + 0.5091*200) = -6.2727 (30j)

140 – (24.4530 + 0.5091*220) = 3.5454 (30k)

155 – (24.4530 + 0.5091*240) = 8.3636 (30l)

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150 – (24.4530 + 0.5091*260) = -6.8181 (30m)

As reflects from the above computations, error term reflects how much an individual Y deviates

from its estimated value. The values of error terms play important role in determining the size of

variance Ϭ2 (computational formula 6), which further affects a number of other computations.

A characteristic of error or residual term is that, once we add or take its mean value, it turns out

equal to zero, in both cases.

1.6 Evaluating the estimated model

After running regression, the results are reported usually reported in the following form.

Y = 24.4530 + 0.5091X (31a) (6.4140) (0.0357) (Standard error) (31b) (3.8124) (14.2445) (t-statistic) (31c)

(0.005) (0.000) (p-value/sig. level) (31d) R= 0.981 R2 = 0.9621 R2

adjusted = 0.957 F = 202.868 (p-value = 0.000) DW = 2.6809 N = 10 (31e)

The econometric model is specified in the form of 1 (a or b), estimated in the form of 31 (a) and

evaluated, using the diagnostic statistic provided in 31(b – e). The estimated model’s evaluation

is carried out, using three distinct criteria, namely:

(a) Economic/management theory criteria (expected signs carrying with the coefficients

of X-variables)

(b) Statistical theory criteria (t statistic or p-value, F statistic, and R2)

(c) Econometrics theory criteria (Autocorrelation, Heteroscadasticity &

Multicollinearity)

Economic theory criteria

Questions:

a) Are these results in accordance with the economic theory?

b) Are they in accordance with our prior expectation?

c) Do the coefficients carry correct sign?

Answer: Yes, we expected a positive relationship between the income of a family and its

consumption expenditure. The coefficient of income variable, X, is positive.

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Statistical theory criteria

Question 1:

a) Are the estimated regression coefficients significant?

b) Are the estimated regression coefficients ßs individually statistically significant?

d) Are the estimated regression coefficients ßs individually statistically different from zero?

Answer: Here, we need to test the hypothesis:

HO: ß1 = 0

H1 : ß1 ≠ 0

= (.5091 – 0) / .0357 = .5091 / .0357 = 14.2605

(32) Our t calculated = 14.2605 > t tabulated = 1.86 at .05 level of significance, with df (N – k) = 8; hence, we

reject the null hypothesis; the coefficient ß1 is statistically significant. Another way of checking

the significance level of ßi coefficients is to check its respective p-value (Sig. level). In case of

the coefficient of X-variable, the p-value = 0.00, suggesting that coefficient ß1 is statistically

significant at p < 0.01. In this second case, we do not need to check the statistical significance

level, using the t-distribution table appended at the end of some econometrics book; we can

directly check p-value provided next to the t-value in the output of the solved problem.

Question 2:

a) Are the estimated regression coefficients collectively significant?

b) Do the data support the hypothesis that

ß1 = ß2 = ß3 = 0

Here, we need to test the hypothesis:

HO: ß1 = ß2 = ß3 = 0

H1: ßi are not equal to 0

Answer: Here, we use F-stattistic, namely:

(33)

= 202.868

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Our F statistic (F = 202.868 > F 1, 8; .05 = 5.32) suggests that the overall model is statistically

significant. Like in case of t-statistics, the significance level of F-statistic can also be checked

from p-value given next to Fcalculated in the output of the solved problem.

Question 3: Does the model give a good fit?

Answer: Yes; our R2 = 0.9621 suggests that 96.21% variation in the dependent variable

(Y) has been explained by variations in explanatory variable (X).

Econometrics theory criteria

1) No Autocorrelation Criteria (We will discuss

2) No Heteroscadasticity Criteria (these criteria in detail

3) No Multicollinearity Criteria (later on in the course

1.7 Interpreting the results of regression analysis

The estimated results suggests that if there is one unit change in explanatory variable X

(family’s income), there will be about half unit (.5091) change in dependent variable Y (family’s

consumption expenditure). If X and Y both are in rupees, then it means that there will be 51

paisas increase in consumption expenditure if the family’s income increases by one rupee.

1.8 Standard assumptions of Least-Square estimation techniques

The linear regression model is based on certain assumptions; if these assumptions are not

fulfilled, then we have certain problems to deal with. These assumptions are:

1. Error term μ i is a random variable, and has a mean value of zero.

===> μ i may assume any (+), (-) or zero value in any one observation/

period, and the value it assume depends on chance.

The mean value of μ i for some particular period, however, is zero, i.e.,

∑ (μ i / xi) = 0

2. The variance of μ I is constant in each period, i.e.,

Var (μ i ) = б2

This is normally referred to as homoscedasticity assumption, and if this

Assumption is violated, then we face the problem of heteroscedasticity.

3. Based on assumption 1 and 2 , we can say that variable μ i has a normal

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distribution, i.e.,

μ i ~ N(0, б2)

4. Error term for one observation is independent of the error term of other

observation, i.e., μ i and μ j are not correlated, or

Cov (μ i and μ j ) = 0

This is no-serial-autocorrelation assumption, and if this assumption is

violated, then we have autocorrelation problem.

5. μ i is independent if the explanatory variables (X), that is, the μ i and μ j are

not correlated.

Cov (X μ ) = ∑{[Xi - ∑ (Xi)] [ μ i -∑ (μ i)]} = 0

6. The explanatory variable (Xi) are not linearly correlated to each other; they

do not affect each other. If this assumption is violated, then we face the

multicolinearity problem.

7. There is no specification problem, that is,

a) Model is specified correctly, mathematically, from the economic

theory point of view.

b) Functional form of the model ( i.e., linear or log-linear or any other

form) is correct.

c) Data on dependent and independent variables have correctly collected,

i.e., there is no measurement error.

1.9 BLUE properties of estimator:

Given the aforementioned assumptions of the classical linear regression model, the Least -

Square estimator (β) possess some ideal properties.

1. It is linear.

2. It is unbiased, i.e., its average or expected value is equal to its true

value.

Biasness can be measured as:

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3. It is minimum- variance, i.e. it has minimum variance in the class of all such Linear

unbiased estimators.

4. It is efficient. An unbiased estimator with the least variance is known as an

Efficient estimator. From properly (2) and (3), our OLS estimator is unbiased and minimum

variance, so it is an efficient estimator.

5. It is BLUE, i.e., Best-linear-unbiased estimator.

There is a famous theorem known as “Gaus-Markov Theorem” which tells:

“Given the assumptions of the classical linear regression model, the least-square

Estimators, in the class of unbiased linear estimators, have minimum variance, So they are

best-linear unbiased estimators, BLUE”.

Assignment 1(Due in the next class)

You have already received Gujarati’s (2007) ‘Basic Econometric’; study its relevant section to solve the following assignment..

1. Study sections 1.4 & 1.5: How does regression differ from correlation?2. Read section 1.6: What are some other names used for dependent and independent

variables? 3. Study section 1.7: What are different types of data? Explain each type in one or two

sentences.4. Study example 6.1 (page 168-169): Which of the two estimated model (6.1.12 & 6.1.13)

is better and why? What do you learn from this example, in general.

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Topic 2

Simple Regression to Multiple Regression Analysis

2.1 Shortcomings of two-variable regression analysis

In spite of providing the base for general regression, the simple or two-variable regression has

certain limitations; it gives biased results (of Least-Square Estimators, βs) if specified model

excludes some relevant explanatory variables (namely X2, X3, …..).

Let’s revisit to our first topic’s example of “Families’ Consumption’, wherein model was

specified and run, as follows.

Y = β0 + β1X + e

= 24.4530 + 0.5091 X (6.4140) (0.0357) (Standard Error) (3.8124) (14.2445) (t-statistic)

(0.005) (0.000) (p-value/sig. level) R= 0.981 R2 = 0.9621 R2

adjusted = 0.957 F = 203.082 (p-value = 0.000) DW = 2.6809 N = 10 (2.1)

If we recall, the results of this estimated model, while we evaluated in terms of economic theory

(sign of the coefficient carrying with X) and statistical theory criteria (t-statistic/p-value, F-

statistic and R2), were turned out to be reasonably acceptable. But, while we reconsider the

specification of the model, we will find that we had misspecified the model at the first place;

according to the theory, consumption (Y) depends on income (X1), as well as, wealth of the

families (X2), prices of consumption items (X3), prices of the related

products/substitutes/complements (X4), and so on. Hence, in spite of the fact that results

provided in (2.1) are apparently seem reasonable in light of the diagnostic statistic used, the

estimated model provides biased results as it does not include some very important and relevant

explanatory variables.

Solution then lies in the Multiple regression analysis, wherein all relevant explanatory variables

need to be included, like the following one.

Y = β0 + β1X1 + β2X2 + β3X3 + …………. + βNXN + e (2.2)

Let’s take a practical example of using multiple regression analysis (see next sub-section 2.2).

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2.2 An example of multiple regression analysisIn case, research topic is:

“Organizational justice and employees’ job satisfaction: a case of Pakistani organizations”

Knowing that ‘organizational justice’ has 4 well identified facets, namely:

1. Distributive justice (JS)

2. Procedural justice (PS)

3. Interactive justice (IJ), and

4. Informational justice (INJ)

Assuming that, if organizational justice prevails in Pakistani organizations, then employees

would be satisfied (job satisfaction, JS); hence, respective econometric model may be specified,

as follows.

JS = f(DJ, PJ, IJ, INJ) (2.3)

We may estimate this model in linear and/or log-linear form, that is:

JS = α0 + α1DJ + α2PJ + α3IJ + α 4INJ + ei (Linear model) (2.4)

lnJB = β0 + β1lnDJ + β2lnPJ + β3lnIJ + β4lnINJ + μi (Log-linear model) (2.5)

(Note: ‘ln’ stands for natural log)

Steps (to be taken):

For estimation of linear model

1. As per requirements of the model specified in (2.3), we need to develop a questionnaire,

like the one placed at Annex – I; and then collect the required data.

2. Enter the data collected on the employees’ responses in SPSS, using data editor

(spreadsheet like that of EXCEL-spreadsheet). Check how data has been entered in file

named: CLASS-EXERCISE-DATA_1.

3. Estimate reliability test (Chronbach’s Alpha) of the raw-data on employees’ responses,

separately for each of the constructs used (JS, DJ, PJ, IJ & INJ).

4. Try to understand what reliability, validity and generalizability concepts stand for (see

Annex – II). Interpret the results of reliability test (See ANNEX – III)

5. Generate data on variables of interest, namely: JS, DJ, PJ, IJ & INJ.

6. Run regression model specified in (2.4), and report the results.

JS = 2.371 + 0.098DJ - 0.021PJ + 0.076IJ + 0.292INJ - 0.005AEE(9.882) (2.199) (-0.509) (1.905) (4.472) (-1.636)(0.000) (0.029) (0.611) (0.058) (0.000) (0.103)

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R= 0.506 R2 = 0.2560 R2adjusted = 0.2410

F = 17.71 (p-value = 0.000) DW = 1.5930 N = 264 (2.6)

(Figures in the first and second parentheses, respectively, are t-statistics and p-values)

Note: AEE stands for the combined figures of age, education and experience of the employees,

and have been included to capture the combined effects of these variables.

For estimation of log-linear model

7. Convert newly generated data on JS, DJ, PJ, IJ & INJ and AEE into their logs

8. Run model 2.5, and report the results

lnJS = 0.943 + 0.156lnDJ - 0.015lnPJ + 0.080lnIJ + 0.308lnINJ - 0.084lnAEE(4.594) (2.829) (-0.308) (1.554) (4.506) (-1.645)(0.000) (0.005) (0.758) (0.122) (0.000) (0.101)

R= 0.522 R2 = 0.2720 R2adjusted = 0.2580

F = 19.309 (p-value = 0.000) DW = 1.618 N = 264 (2.7)

Evaluation and interpretation of the estimated models

Linear model 2.6

(a) Model is found statistically significant (F = 17.71, p < 0.01); though all the

explanatory variables included in the model seem to have explained around 25

percent variance in the dependent variable (R2 = 0.2560; R2adjusted = 0.2410).

(b) Variable PJ appears to be highly statistically insignificant (p = 0.611), compared to

variables INJ and DJ with highly statistically significant contribution (p < 0.01 & p <

0.05 ) and variable IJ and AEE with moderately statistically significant contribution

(p = 0.058 & p = 0.103).

(c) Results suggest that variables INJ, DJ and IJ positively contribute towards

determination of employees’ job satisfaction, AEE negatively contributes while PJ

does not contribute. The negative relationship of AEE with JB suggests that

employees of higher age, with relatively higher education and experience, are less

satisfied from their jobs.

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Log-linear model 2.7

(a) Since the two formulations of the data (nominal-data and log-data), used in linear and

log-linear models, differ from each other, we cannot compare results of one model

with that of the other. However, we expect relatively better results from a log-linear

model; so we can discuss whether or not the results have been improved. Yes, results

are relatively improved, especially in terms of F-statistic and t-statistic/p-values.

Model is found statistically significant (F = 19.309, p < 0.01); the explanatory

variables explain around 27 percent variance in the dependent variable (R2 = 0.2720;

R2adjusted = 0.2580).

(b) Log-linear model reinforces the results regarding signs and significance values of the

individual explanatory variables.

(c) Results (of the both models) suggest that facets like informational justice, distributive

justice and informational justice appear to be positively contributing towards

employees job satisfaction, as compared to the procedural justice, which needs to be

taken care of for an overall satisfaction of Pakistani organizational employees. In

addition, the senior, more educated and more experienced employees also need

attention as they appear to be mostly dissatisfied from their jobs.

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Assignment 2 (Due in the Next Class)

1. Briefly explain (in bullet-points) what the major contribution is that of simple/two-variables regression model, and why we have to resort to multiple regression analysis.

2. Go through the steps suggested for estimation of a linear-regression model; what is the difference between a linear and log-linear model? (a) How do the steps of estimation of a log-linear model differ from that of linear model? (b) How do the interpretations of the two model differ?

3. What is reliability? How is reliability test run in SPSS? Why is the running of reliability test important?

4. What is the procedure of generating data on variables of interest? How is a Likert-scale questionnaire used for generation of data on variables of interest?

5. How are and for what purposes, F-statistic, R2 and t-statistic/p-values used for the evaluation and interpretation of estimated models?

6. Study material (entitled “Formulating and clarifying a research topic”) provided in Annex – IV:(a) In Part – I (of Annex – IV), the answers of the following two questions have been

provided:1. What are three major attributes of a good research topic?

2. How can we turn research ideas into research projects?

(b) In Part – II, you have been provided two lengthy lists of research topics proposed by my MS ARM’s class students of section 2 & 3. You please select one topic of your choice (select topic in light of what you have learnt from materials provided in Part – I), develop 2 – 3 research questions and 4 – 5 research objectives, and submit me through email ([email protected] & [email protected]), latest by 12.00 (Noon) Monday; please note: we will discuss your selected topic along with research questions and objectives in Monday’s evening class (along with the remaining/leftover part of previous Lecture – 2). Please also note: you may suggest a topic of your own (not already enlisted), along with research questions and objectives. Whether you select a topic from our list or suggest the one from your own side, two students of my ARM class will assist you to carry out research on that topic, as part of your AQT class requirements, for a 20% marks.

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ANNEX – I (Questionnaire)

Section IYour Organization (Tick 1 or zero): Government = 1 2. Private = 0Your gender (Tick 1 or zero): Male = 1 2. Female = 0Your age (in years like 25 years, 29 years,)Your education (actual total years of schooling, like 14 years; 18 years)Your area of specialization:Your job title in this organization:Experience: Working years in this organization:

Section II

Strongly disagree – 1 Disagree = 2 Not disagree/neither agreed = 3 Agreed = 4 Strongly agreed = 5

JS: Job satisfaction (Agho et al. 1993; Aryee, Fields & Luk (1999)) 1 2 3 4 5

1 I am often bored with my job (R)

2 I am fairly well satisfied with my present job

3 I am satisfied with my job for the time being

4 Most of the day, I am enthusiastic about my job

5 I like my job better than the average worker does

6 I find real enjoyment in my work

Organizational Justice (Niehoff and Moorman (1993))Strongly disagreed = 1 Slightly disagree = 2 Disagree = 3 Neutral (Not disagree/neither

agreed) = 4 Agreed = 5 Slightly more agreed = 6 Strongly agreed = 7Distributive justice items (DJ) 1 2 3 4 5 6 7

1 My work schedule is fair

2 I think that my level of pay is fair

3 I consider my workload to be quite fair

4 Overall, the rewards I receive here are quite fair

5 I feel that my job responsibilities are fair

Procedural justice items (PJ) 1 2 3 4 5 6 7

1 Job decisions are made by my supervisor in an unbiased manner

2 My supervisor makes sure that all employee concerns are heard before job decisions are made

3 To make formal job decisions, supervisor collects accurate & complete information

4 My supervisor clarifies decisions and provides additional information when requested by employees

5 All job decisions are applied consistently across all affected employees

6 Employees are allowed to challenge or appeal job decisions made by the supervisor

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Interactive justice items (IJ)1 When decisions are made about my job, the supervisor treats me with

kindness and consideration 2 When decisions are made about my job, the supervisor treats me with

respect & dignity3 When decisions are made about my job, supervisor is sensitive to my

own needs4 When decisions are made about my job, the supervisor deals with me

in truthful manner5 When decisions are made about my job, the supervisor shows concern

for my rights as an employee6 Concerning decisions about my job, the supervisor discusses the

implications of the decisions with me7 My supervisor offers adequate justification for decisions made about

my job8 When decisions are made about my job, the supervisor offers

explanations that make sense to me9 My supervisor explains very clearly any decision made about my job

Strongly disagree – 1 Disagree = 2 Not disagree/neither agreed = 3 Agreed = 4 Strongly agreed = 5Informational justice items (INJ) 1 2 3 4 5

1 Your supervisor has been open in his/her communications with you

2 Your supervisor has explained the procedures thoroughly

3 Your supervisor explanations regarding the procedures are reasonable

4 Your supervisor has communicated details in a timely manner

5 Your supervisor has seemed to tailor (his/her) communications to individuals’ specific needs.

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ANNEX - IICredibility of research findings: important considerations

(Reliability? Validity? Generalizability?)

Reliability: Reliability can be assessed by posing three questions:

1. Will the measure yield the same results on other occasions?

2. Will similar observations be reached by other observers?

3. Is the measure/instrument stable and consistent across time and space in yielding

findings?

4-Threats to reliability

(i) Subject/participant error

(ii) Subject/participant bias

(iii) Observer error and

(iv) Observer’s bias

Validity: Whether the findings are really about what they appear to be about.

Validity depends upon:

History (same history or not),

Testing (if respondents know they are being tested),

Mortality (participants’ dropping out),

Maturation (tiring up), and

Ambiguity (about causal direction).

Generalizability:

The extent to which research results are generalizable.

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ANNEX – III

Reliability test and interpretation

Reliability test results

Responses on the elements of all five constructs (JS, DJ, PJ, Ij & INJ) were entered on SPSS’s

data editor and reliability tests were conducted; the following Cronbach’s Alphas were

estimated.

Table 4.4 Results of reliability testConstruct Cronbach’s AlphaJob Satisfaction (JS) 0.739Distributive Justice (DJ) 0.828Procedural Justice (PJ) 0.890Interactional Justice (IJ) 0.920Informational Justice (INJ) 0.834

InterpretationAccording to Uma Sekaran (2003), the closer the reliability coefficient Cronbach’s Alpha gets to

1.0, the better is the reliability. In general, reliability less than 0.60 is considered to be poor, that

in the 0.70 range, acceptable, and that over 0.80 and 0.90 are good and very good. The reliability

tests of our constructs happened to be in the acceptable to good and very good ranges.

23


ANNEX - IVFormulating and clarifying a research topic1

Part – I: Two major questions:

3. What are three major attributes of a good research topic?

4. How we can turn research ideas into research projects

Three major attributes of a good research topic are

Is it feasible?

Is it worthwhile?

Is it relevant?

Capability: is it feasible?» Are you fascinated by the topic?» Do you have the necessary research skills?» Can you complete the project in the time available?» Will the research still be current when you finish?» Do you have sufficient financial and other resources?» Will you be able to gain access to data?

Appropriateness: is it worthwhile?

» Will the examining institute's standards be met?» Does the topic contain issues with clear links to theory?» Are the research questions and objectives clearly stated?» Will the proposed research provide fresh insights into the topic?» Are the findings likely to be symmetrical?» Does the research topic match your career goals?

Relevancy: is it relevant?» Does the topic relate clearly to an idea you were given - possibly by your organisation?

Turning research ideas into research projects

Conceive some research idea

Think about research topic (having attributes stated above)

Write research questions

Develop research objectives

1 This discussion is based on materials contained in chapter 2 of Saunders, M., Lewis, P. and Thornhill, A. (2011) Research Methods for Business Students 5th Edition. Pearson Education

24


Part – II: Research topics proposed by MS-ARM students

ARM (section – 2)

Performance appraisal as a tool to motivate employees: a comparison of public-private sector

organization

Performance appraisal in ……………….. (name of organization)

Marketing communication and brand loyalty

Implementation of Integrated Management System (IMS) in Pakistan Civil Aviation Authority

Information technology and financial services

Capital structure and firms profitability

Interest rates, imports, exports and GDP

Intra-Group Conflict and Group Performance

HR practices across public and private organizations

HR practices across SMEs and large companies

HR practices across manufacturing and services sector companies

Corporate governance practices in banking sector of Pakistan

Corporate governance practices in textile industry

Corporate governance practices in pharmaceutical industry

Effects of working capital management on profitability

Working capital with relationship to size of firm

Working capital and capital structure

Optimizing working capital

Dividend policy and stock prices

Sales, debt-to-equity ratio and cash flows

Relationship between KSE’s, LSE’s and ISE’s stock prices

Gold prices and stock exchange indices

Interest rates, bank deposits and private investments

Security Market Line (SML) & Capital Market Line (CML) at KSE

Relationship between stock market returns and rate of inflation

Relationship between CPI and Bond price

25


Pakistan’s exchange rates with relation to major global currency regimes: an analysis

ARM (section – 3)

Trade deficit, budget deficit and national income

Performance appraisal and its outcomes

Impact of compensation on employee’s job satisfaction

Human resource management & outsourcing

Advertising and brand image

Performance management in public sector organizations

Impact of training on employees’ motivation and retention

Impact of performance appraisal

Financial returns, returns on shares, equity returns and share prices

Factors contributing towards employee turnover intention

Antecedents of employees’ retention

Employees’ retention policies and employees’ turnover

Impact of training and development on employees’ motivation and turnover intention

Outsourcing human resource function in Pakistani organizations

Exploring the impact of human resources management on employees’ performance

Service orientation, job satisfaction and intention to quit

Brand equity and customer loyalty: a case of …….. (name of orhanization)

PTCL privatization: effects on employees’ morale

PTCL privatization: effects on employees’ efficiency

PTCL privatization: effects in terms of profitability

Electronic and traditional banking: how do customers’ perceive?

FPI and FDI in Pakistan: a comparative analysis

Stock market indices: KSE, LSE and ISE compared

Work family conflict and employee job satisfaction: moderating role of supervisor’s support

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Topic 3

Multiple regression: model specification

3.1(a) Conceiving research ideas and converting it into research projects: aProcedure

Procedure: Research ideas à research topic à research questions à research

Objectives à research hypotheses

Your Take-home Assignment 2’s question 6 has set the example how research ideas and topics

are converted in to research projects, adopting the procedure detailed above. Students have also

provided details of their chosen topics; let’s discuss those topics and clarify them further,

judging them in light of the relevant theories (section 3.1b).

3.1(b) Incorporating theory as the base of your research

Econometrics theory

Please study section 7.2 and 7.3 of Andren (2007)2 and try to understand what difference it

creates when we omit a relevant explanatory variable or include an irrelevant one in an

econometrics model.

Economics/management theory

Let us evaluate whether the research projects you have proposed are based on the relevant

economic/management theory, and if not, then how you can incorporate the relevant theory into

your projects.

Discussion on your proposed research projects

(You need to take notes on suggestions for improvements, and submit theimproved version of your research project as part of your next assignment 3 (a).

(See Annexure – I for topics for discussion

Assignment 3 (a)

1. You must have taken the notes on suggestions made during our class discussion on your respective research projects; you please refine your topics and research questions and objectives, in light of the discussions as well as what the following research articles suggest

2 Andren, Thomas. (2007). Econometrics. Bookboon.com, pp.74-77

27


regarding basing your research on relevant theory (soft copies of papers are provided on AQT-Class Yahoo Group).

Article/Note: ‘Formulating a Research Question’

Rogelberg, Adelman & Askay (2009). Crafting a Successful Manuscript: Lessons from 131 Reviews. J Bus Psychol (2009) 24:117–121 (Study only 8-points given under heading ‘Conceptual and/or theoretical rationale’.)

Thomas, Cuervo-Cazurra & Brannen (2009). From the Editors: Explaining theoretical relationships in international business research: Focusing on the arrows, NOT the boxes. Journal of International Business Studies (2011) 42, 1073–1078 (Read only ‘Abstract’ and ‘Introduction’ sections, and try to understand Figure 1 (Typical conceptual diagram).

Andren, Thomas. (2007). Econometrics. Bookboon.com (Read only sections 72 & 73,

pp.74-77)

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Topic 3 Multiple regression: model specification….continues

In sub-section 3.1(a), we carried out an exercise on how a conceived research idea can be

converted in to a research projects (Research ideas à research topic à research questions à

research objectives). In sub-section 3.1(b), we tried to learn how much important the

econometrics (omission and inclusion of relevant and irrelevant explanatory variables) and

economics/management theories are for specification of an econometrics model. In this new

subsection 3.2, we will try to learn what role different mathematical formulations can play in

econometrics modeling

3.2 Specifying an Econometric Model: Mathematical Specification

This section further consists of two subsections, namely:

3.2(a) Specification of an econometric model: mathematical formulation in general

3.2(b) Some practical examples of mathematical formulations/specifications: production

function, cost-function and revenue function

3.2(a) Specification of an econometric model: mathematical formulation in general

Our discussion in earlier sections on simple regression and multiple regression analysis clarifies

two major points, namely:

1. The simple and multiple regression analysis assumes that variable Y depends on variable

X, but for this phenomenon of dependence or causation, the researcher takes insights

from the basic theory (economics/management).

2. Previous discussion further emphasizes that it is the researcher’s responsibility to specify

an econometric model such that it contains all major relevant explanatory variables as

independent variables; otherwise, empirical results obtained in terms of estimated

coefficients would be biased.

While specifying a model, the researcher has to take the above points in to consideration.

Additionally, the researcher has to decide which mathematical formulation of the model he/she

should use so that the true relationship between dependent and independent variables is captured

to the maximum extent. This is how an econometric model is/should be specified.

29


Let’s proceed further, taking some practical examples of mathematical formulations of the

model. In case, we have the following type of relationship between Y – X variables:

Y Y Y

X X XCase 1 (a) Case 1 (b) Case 1 (c)

Case 1a is a general linear relationship, and can be measured, as follows.

Y = β0 + β1X1 + e (3.1)

In 3.1, we expect β1to carry positive sign.

The case 1(b) represents an exponential case, and can be measured, as follows:

Y = β0 + β1X1 + β2X21 + e (3.2)

Specially, the parameters β1and β2 will carry positive signs.

In case of a cubic-type of relationship like 1(c), the following mathematical formulation will have to be

adopted.

Y = β0 + β1X1 + β2X21 + β3X3

1 + e (3.3)

The coefficients β1and β2 will carry positive but β3 negative sign.

In other words, it means that if we have to measure the stated type of relationships between

our Y – X variables, we need to use the relevant type of mathematical formulations while

specifying our econometrics model.

In certain other cases/on certain occasions, we have to adopt some other mathematical

formulations like the following ones:

Y = β0 + β1X1 + β2X1X2 + β3X2 + e (3.4)

Y = β0 + β1X1 + β2X21 + β3X1X2 + β4X2 + β5X2

2 + e (3.5)

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Equation 3.4 measures linear relationship, but includes an interaction term (X1X2). β2 can take

any sign (+, - or 0); a positive sign would show positive effect of the interaction of X1 and X2 on Y, a

negative sign would mean negative effect of interaction of these two variables and zero effect

would mean zero effect on dependent variable Y. Let’s visit some practical examples where we

can use some of the above stated mathematical formulations (next section).

3.2(b) Some practical examples: production, cost and revenue functions

Production functionIn case, we have data on production of product Y, wherein two major inputs used are X1 and X2:

Y X1 X2

2500 1 1502525 2 1522555 3 1552592 4 1592635 5 1612677 6 1692718 7 1742745 8 1782766 9 1812781 10 182

Let’s check relationship between Y – X1, and Y – X2 (separately), using mathematical formulation given

in (3.3), using data provided in above table.

Do this as Take-home Assignment 3b (Question 1); show the estimated

relationship through hand-drawn graph

Let’s check relationship between Y and X1 & X2, using mathematical formulation given in (3.4), using

data provided in the above table.

Do this as Take-home Assignment 3b (Question 2); interpret the results,

including that of the interaction term

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Cost Function

Cost function can be developed when you have data like the following one:

Y TC1 1932 2263 2404 2445 2576 2607 2748 2979 350

10 420

Mathematical formulation of a typical cost-function is:

TC = β0 + β1Y - β1Y2 + β1Y3 + e (3.6)

Did you notice the signs of a typical cost-function are opposite to that of a typical production-function

(given in 3.3).

Estimate cost-function 3.6 as Take-home Assignment 3b (Question 3); show

the estimated relationship through hand-drawn graph

Assignment 3b: Question 4Download 8 – 10 published research articles on the area of

research/topic you have chosen for your class research project, study the conceptual models tried in these research articles, and

develop your own model, including the mathematical one as part of your Take-home Assignment 3(b), due in next class; be ready for a

class presentation also.

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Topic 3 Multiple regression: model specification….continues

3.3 Conceptual/econometric modeling

3.3 (a) Examples in Finance

3.3 (b) Examples in Marketing

3.3 (c) Examples in HRM

3.3 (a) Examples in Finance: summary

Example 1: Interest rates and GDP: a case of Pakistan

Example 2: Capturing effects of interest rates on Pakistani economy

Example 3: Exchange rates and Pakistan’s trade: an analysis

Example 4: Exchange rates and Pakistan’s economy: an analysis

Example 5: Research on Working Capital (WC)

Proposal 1: “Relationship between Profitability and Working CapitalManagement”, using econometric technique

Proposal 2: “Liquidity-profitability trade-off”, using Goal programming (ofOperations Research)

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3.3 (a) Examples in Finance

Example 1: Interest rates and GDP: a case of Pakistan3

Though we are interested in analyzing the effect of interest rates on Pakistan’s national income,

but we know that interest rates do not affect GDP directly, rather these affect saving (bank

deposits) and private investments, and as a consequence GDP is affected; so we conceptualize

the path of the effect, as follows:

Interest rates (↑↓) à bank deposits (↑↓) & private investments (↓↑)

à GDP (↓↑)

The above path of the effect (of interest rates) can be captured, through econometrics model,

postulated, as follows.

Private investment = ƒ(Interest rates) (3.7a)

GDP = ƒ(Private investments_predicted in equation 7a) (3.7b)

Theory tells us that private investment (PI) is influenced not only by the interest rate (R) but is

also affected by openness of the economy (OE) and, especially the costs and taxes (C&T).

Hence, equation 3.7a would change to:

PI = ƒ(R, OE, C&T) (3.8a)

The private investment predicted on the basis of equation 3.8a (PI) is not the only determinant of

GDP, government expenditure (GE) or budget spending is another determining variable; while in

Pakistani context, Foreign Direct Investment (FDI) and Pakistan’s productive population, that is,

the active labor force (LF) are two other factors should be considered as determinants of

Pakistan’s national income (GDP). Hence, model 3.7b would change, as follows.

GDP = ƒ(PI, GE, FDI, LF) (3.8b)

The model postulated in 3.8 (a – b) still needs improvement; government expenditure (GE) and

FDI are not autonomous in nature, the former depends on government revenues (GR) and

government borrowing from foreign (FB) and domestic (DB) sources, and the latter depends

3 Students are urged to think over the difference between topic of this Example 1 and that of Example 2, and then try to understand how conceptual/econometric modeling can be differently developed to take care of the differences which the two topics necessitate.

34


upon economy’s openness (OE) and cost of production and taxes (C&P). To incorporate these

effects, the model would therefore adopt the following form.

PI = ƒ(R, OE, C&P) (3.9a)

GE = ƒ(GR, FB, DB) (3.9b)

FDI = ƒ(OE, C&P) (3.9c)

GDP = ƒ(PI, GEi, FDI, LF) (3.9d)

Model 3.9 (a – d) represents what we need to do for a piece of research conducted under title

“Interest rates and GDP: a case of Pakistan”. In case we extend the scope of our research to what

is needed under title “Capturing effects of interest rates on Pakistani economy”, we will then

have to adopt the model specified in the following Example 2.

Example 2: Capturing effects of interest rates on Pakistani economy

Notice the difference between the two topics (Example 1 and 2); the first topic requires

analyzing the effect of exchange rates on GDP, while the second topic asks for looking in to the

same thing from a little broader perspective, that is, from the point of view of whole economy.

Since the model specified for the first topic covers largely the methodology needed for the

second topic, we can use the same first example model 3.9 (a – d), with an additional equation

for analyzing the effect of interest rates on bank deposits, which can be assumed to be

determined by money supply in the country (M), in addition to the interest rates (R).

Bank deposit = ƒ(R, M) (3.9e)

Hence, model 3.9 (a – e) will be used for the piece of research identified in example 2.

Example 3: Exchange rates and Pakistan’s trade: an analysis4

According to the theory, the appreciation or depreciation of exchange rates (ER) affects the

country’s trade; appreciation of a country’s currency makes exports expensive and imports

cheap, and depreciation makes exports cheap and imports expensive. This stated phenomenon is

true for the two trade partners, but is also affected by certain other situations prevailing in the

two trading countries. The foreign country’s exchange rates with respect to her other major trade

4 Students are urged to think over the difference between topic of this Example 3 and that of Example 4, and then try to understand how conceptual/econometric modeling can be differently developed to take in to account the differences which the two topics necessitate.

35


partners, availability and prices of the substitutes in foreign country and world over, consumers’

income, trade openness and political situations are some other important factors affecting export

and import trade.

Tracing and finding out the effects of the determinants of export and import trade might be easy

when trade of certain known commodities between two specific countries is analyzed; but the

case becomes cumbersome, and needs extra care when analysis of trade is required at aggregate

level, for instance the topic of this piece of research - Exchange rates and Pakistan’s trade: an

analysis.

We can think primarily about some very simple questions like what the exchange rates are

(definition), how these are determined (or are autonomous in nature), they affect what and how,

and specifically what relationship they have with trade – its two components, imports and

exports. And since we are analyzing the exchange rates of Pakistan and her trade, we should

think over the answers of such questions in the context of Pakistan’s economy.

Exchange rates (ER) are not autonomous in nature, these are determined by the forces of demand

for and supply of major medium of currency (US dollar in Pakistan) used in imports and exports

trade. Value of imports seems to be the major factor to determine demand for US dollar in

Pakistan, and while value of exports, workers’ remittances (WR), foreign direct investment

(FDI) and foreign borrowings (FB) appear to be the major determinants of supply of dollar.

Hence, these demand and supply factors determine exchange rates in Pakistan, which in turn

affect volumes of import and export.

ER = ƒ(IM, EX, WR, FDI, FB) (3.10)

IM = ƒ(ERi ) (3.11)

EX = ƒ(ERi ) (3.12)

But ERi is not the only determinant of import (IM). Imports in Pakistan have historically been

largely composed of capital goods (28% in 1980-81 and 24% in 2010-11) and industrial raw

materials (58% in 1980-81 and 60% in 2010-11)5; the value of the share of Pakistan GDP’s

manufacturing sector (GDPM) may therefore be included in equation 3.11 as proxy to represent

the demand for imports, in addition to the population or its growth rate (POP) as proxy for the

size of the market. Hence, equation 3.11 adopts new form, namely:

IM = ƒ(ERi , GDPM, POP) (3.13)

5 Government of Pakistan (2012). Pakistan Economic Survey 2011-12. Statistical Appendix Table 8.5B

36


In case of exports, primary commodities and semi-manufactured and manufactured products

have been the major components, with share of 44% in 1980-81 and 18% in 2010-11, 11% in

1980-81 and 13% in 2010-11 and 45% in 1980-81 and 69% in 2010-11, respectively6. The values

of the primary (GDPP) and secondary/manufacturing sectors’ contributions to GDP (GDPM)

may therefore be included in equation 3.12 as proxies to represent major supplying sectors of

exports. The demand for Pakistani exports has come from both developed (60.8% in 1990-91 and

44.5% in 2010-11) and developing (39.2% in 190-91 and 55.5% in 2010-11) countries7, the

world’s GDP can be taken as proxy to represent demand from the whole world (GDPW). Hence,

equation 3.12 adopts the new form, namely:

EX = ƒ(ERi , GDPP, GDPM, GDPW) (3.14)

Summarizing the model,

ER = ƒ(IM, EX, WR, FDI, FB) (3.15a)

IM = ƒ(ERi , GDPM, POP) (3.15b)

EX = ƒ(ERi , GDPP, GDPM, GDPW) (3.15c)

We can add even some other relevant variables and improve the model (model 3.15), and

reviewing the relevant literature on respective topics and sub-topics, with special reference to

Pakistan, would help us in this regards.

Please note that model 15 (a – c) will restrict research to the analysis of the effects of exchange

rates on Pakistan’s trade; in case, if someone is interested to analyze the exchange rates’ effects

on Pakistan economy (or GDP), then model specified in following Example 4 should be used.

Example 4: Exchange rates and Pakistan’s economy: an analysis

Model specified in 3.15 (a – c) will work as the base to analyze the effect of exchange rates on

import and export trade, and incorporation of an additional equation (3.15d), which transfers the

effects of imports (IMi ) and exports (EXi ) to GDP will help complete a model for the analysis

necessary for new topic.

GDP = ƒ (IMi , EXi , POP) (3.15d)

The effect of the size of population (POP) has been included as a proxy for the effect of domestic

consumption on country’s GDP.

6 Government of Pakistan (2012). Pakistan Economic Survey 2011-12. Table 8.5A7 Government of Pakistan (2012). Pakistan Economic Survey 2011-12. Table 8.7

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Example 5: Research on Working Capital (WC)

Working capital: in general

Working capital is defined as8:

Working Capital (WC) = current assets (CA) - current liabilities (CL)

(3.16a) Where Current assets are cash and other assets that can be converted to cash

within a year, and Current liabilities are obligations that the company plans to pay off within the year.

Working capital indicates the assets the company has at its disposal for current expenses. The

process of managing the WC efficiently is called Working capital Management. An excess of

working capital many mean that the company is not managing its assets efficiently. It's not using its

assets to get a bigger return or better profit. An aggressive company may keep its working capital

smaller. But a very low working capital may mean the company may not be suited well enough to

payoff its short term obligations.

This decision of how to manage the working capital of the company depends on the Working

capital policy of the company. An important factor that determines the policy is the industry in which

the company operates. For Example, an IT service company may not have a lot of shot-debt in

terms of inventory but it still needs to pay wages, insurances and other expenses like rent. The

company needs to have a policy that makes sure it sets targets were it gets paid as the project

progresses so it can keep paying its staff in time. The company has to manage its account

receivables according to this policy. Some industries operate in a high profit margin that they can

afford to have a longer term on the account receivables because the higher cash balance part of the

current assets. The Collection Ratio helps project this aspect of a company; The collection ratio is

defined as:

Collection Ratio = Accounts Receivable / (Revenue/ 365) 3.16b)

Collection ratio tells us the average number of days it takes a company to collect unpaid invoices. A

ratio which is very near to 30 days is very good since it means that the company is getting paid on a

monthly basis.

Sales is another attribute that strongly impacts working capital. It is the ability of a company to sell its

products fast enough to get the money back to put back into operations or supplies for producing

more materials. Moving inventory fast is always a good plan for a company. It also helps in reducing

costs associated with holding and moving inventory. A good ratio that helps put the attribute in

perspective is inventory turnover ratio, which is defined as:

Inventory turnover ratio = sales / inventory

Or Inventory turnover ratio = Cost of goods sold / inventory (3.16c)

8 The following material is based on http://www.business.com/finance/working-capital/; downloaded on October 12, 2012.

38

http://www.business.com/finance/working-capital/


This ratio shows the efficiency the company has in selling its products. The higher the ratio the better

the company is able to move the products. Again this could be dictated by the industry, for example,

a daily products company is usually forced to sell its products fast enough or lose it. The ratio also

provides a good insight into how a company is doing within an industry. The direct ratio of

companies can be compared to see how well the company is able to sell the products in comparison

to its competitors.

Financing is another attribute of Working Capital management. Debt - Asset ratio provides a good

insight into how much of the company's assets are being financed though debt. The debt – asset

ratio is defuned as:

Debt-asset ratio = Total liabilities / Total assets (3.16d)

Working capital management becomes a very important aspect for a company since it is the first line

of defense against market downturn cycles and recession. A company with cash is usually in a good

position to make better use of the opportunities the markets provide. Its can spend the money on

R&D for coming up with better products. Increase in current assets, especially, increase in account

receivables due to growth is sales have to be managed efficiently. Ability to control working capital

plays a significant role in the survival of the company.

Research on Working Capital

Let us see how the above information on working capital (WC) and working capital management

(WCM) has been used by different researchers to carry out research on the topic under study.

Lazaridis and Tryfonidis’s (2006)9 and Gill, Biger and Mathur (2010)10 analyzed the relationship

between profitability and working capital management, using about the same model, and

measuring and generating the dependent and independent variables in the following way:

No. of Days A/R = (Accounts Receivables/Sales) x 365No. of Days A/P = (Accounts Payables/Cost of Goods Sold) x 365No. of Days Inventory = (Inventory/Cost of Goods Sold) x 365Cash Conversion Cycle = (No. of Days A/R + No. of Days Inventory) – No. of Days A/PFirm Size = Natural Logarithm of SalesFinancial Debt Ratio = (Short-Term Loans + Long-Term Loans)/Total AssetsFixed Financial Asset Ratio = Fixed Financial Assets/Total assetsProfit = (Sales - Cost of Goods Sold) / (Total Assets - Financial Assets)

9 Lazaridis I, and Tryfonidis D, (2006). Relationship between working capital management and profitability of listed companies in the Athens stock exchange. Journal of Financial Management and Analysis, 19: 26-25.10 Gill, A., Biger, N. and Mathur, N. (2010). The Relationship Between Working Capital Management And Profitability: Evidence From The United States. Business and Economics Journal, Volume 2010: BEJ-10

39


Raheman A. and Nasr, M. (2007)11 used similar methodology but measured the required

variables in somewhat different way, namely:

NOPit = β0 + β1(ACPit) + β2 (ITIDit) + β3 (APPit) + β4(CCCit) + β5(CRit) +

β6(DRit)

+ β7(LOSit) + β8(FATAit) + ε

(3.17)

Where:

NOP : Net Operating Profitability

ACP : Average Collection Period

ITID : Inventory Turnover in Days’

APP : Average Payment Period

CCC : Cash Conversion Cycle

CR : Current Ratio

DR : Debt Ratio

LOS : Natural logarithm of Sales

FATA: Financial Assets to Total Assets

ε : The error term.

Researchers have estimated/generated variables, using the following definitions.

Net Operating Profitability (NOP) which is a measure of Profitability of the

firm is used as dependant variable. It is defined as Operating Income plus

depreciation, and divided by total assets minus financial assets.

Average Collection Period (ACP) used as proxy for the Collection Policy is an

independent variable. It is calculated by dividing account receivable by sales

and multiplying the result by 365 (number of days in a year).

Inventory turnover in days (ITID) used as proxy for the Inventory Policy is

also an independent variable. It is calculated by dividing inventory by cost of

goods sold and multiplying with 365 days.

Average Payment Period (APP) used as proxy for the Payment Policy is also

an independent variable. It is calculated by dividing accounts payable by

purchases and multiplying the result by 365.

11 Raheman A. and Nasr, M. (2007). Working capital management and profitability – case of Pakistani firms. International Review of Business Research Papers, 3: 279-300.

40


The Cash Conversion Cycle (CCC) used as a comprehensive measure of

working capital management is another independent variable, and is measured

by adding Average Collection Period with Inventory Turnover in Days and

deducting Average Payment Period.

Current Ratio (CR) which is a traditional measure of liquidity is calculated by

dividing

current assets by current liabilities.

In addition, Size (Natural logarithm of Sales (LOS)), Debt Ratio (DR) used as

proxy for

Leverage and is calculated by dividing Total Debt by Total Assets, and ratio of

financial

assets to total assets (FATA) are included as control variables.

Proposed research (on WC and WCM)

Proposal 1: “Relationship between Profitability and Working Capital Management”,

using econometric technique

Students may use the above reported three studies as guidelines for their own study on

“Relationship between Profitability and Working Capital Management”, using econometric

technique.

Proposal 2: “Liquidity-profitability trade-off”, using Goal programming (of

Operations Research)

About half of our present PhD class students and a good teachers (who have already completed

their PhD course work) have already taken Operations research (OR) course last semester. Let us

see who dare to take the initiative of doing research, using goal programming technique of

Operations research. A good guide in this respect is: Dash, M. and Hanuman, R. A liquidity-

profitability trade-off model for working capital management: electronic copy available at:

http://ssrn.com/abstract=1408722.

Take-home Assignment 3(c)

Q.1 Go through examples 1 and 2, and explain what the difference is in the two topics and how the difference has been taken in to account while postulating the econometrics model.

41

http://ssrn.com/abstract=1408722


Q.2 Go through examples 3 and 4, and explain what the difference is in the two topics and how the difference has been taken care of while postulating the econometrics model.

Q.3 Go through material provided in example 5, and explain what specifically the econometric model 3.17 would be measuring.

42


3.3 (b) Examples in Marketing

MARKETING STUDY 1

How relationship age moderates loyalty formation: The increasing effect of relational equity on customer loyalty.

Maria Antonietta RaimondoUniversità della Calabria, Campus of Arcavacata - Italy

Gaetano “Nino” Miceli Università della Calabria, Campus of Arcavacata - Italy

Michele CostabileUniversità della Calabria, Campus of Arcavacata - Italy

SDA Bocconi Graduate School of Management, Milan - ItalyLuiss Management, Rome - Italy

43

FIGURE 1

Customer Loyalty

Trust

RelationalEquity

Customer Satisfaction

Relationship Age

Attitudinal Loyalty

Behavioural Loyalty


H1: Relational equity has a positive influence on a) attitudinal loyalty and b) behavioural loyalty.

H2: The effects of relational equity on a) attitudinal loyalty and b) behavioural loyalty increase along with the relationship age.

H3: Satisfaction has a positive influence on a) attitudinal loyalty and b) behavioural loyalty.

H4: The effects of satisfaction on a) attitudinal loyalty and b) behavioural loyalty decrease along with the relationship age.

H5: Trust has a positive influence on a) attitudinal loyalty and b) behavioural loyalty.H6: The effects of trust on a) attitudinal loyalty and b) behavioural loyalty increase

along with the relationship age.

Item Mean S.D.Standardized

LoadingConstruct AVE

Composite reliability

Attitude toward focal provider: ability to match customers’ needs

4.35 1.09 .56

Attitudinal Loyalty

.53 .84

Attitude toward focal provider: new value added services

4.43 1.14 .50

Attitude toward focal provider: customer care

4.52 1.12 .73

Attitude toward focal provider: clarity of communication

4.49 1.13 .87

Attitude toward focal provider: completeness of offering and communication

4.45 1.09 88

Positive word-of-mouth 4.70 1.32 .85 Behavioural Loyalty

.68 .81Repurchase intentions 4.80 1.28 .80Overall relationship equity 4.18 1.39 .82

Relational Equity

.54 .85

How fair own benefits relative to own costs

4.18 1.25 .82

How fair own benefits relative to provider’s benefits

3.79 1.44 .65

How fair own benefits relative to provider’s costs

4.19 1.20 .64

Proportionality of customer and provider benefits

4.02 1.27 .73

Overall satisfaction * 4.86 1.00 --

Satisfaction .57 .80Displeased vs. Pleased 4.77 1.04 .72Discontent vs. Content 4.32 1.13 .79Sad vs. Happy 4.46 1.04 .75Service always how I expect 4.18 1.18 .66 Trust .64 .87Reliable provider 5.00 1.20 .82Provider keeps promises 4.66 1.28 .79

44


Trustworthy provider 4.88 1.17 .89

45


MARKETING STUDY 2

The Effect of Marketing Communications and Price Promotion to Brand Equity

Melinda Amaretta † and Evelyn HendrianaHypotheses:

H1: perceived advertising spending has positive effect on perceived quality H2: perceived advertising spending has positive effect on brand awareness H3: perceived advertising spending has positive effect on brand image H4: perceived advertising spending has positive effect on brand loyaltyH5: the use of price deals has negative effect on perceived quality H6: the use of price deals has negative effect on brand image

Research model

Figure 1. The effect of marketing communication on dimensions of brand equity

46


3.3 (c) Examples in HRM

Adopting, adapting or developing a new questionnaire

Example 1: research on‘Job Satisfaction’ versus ‘HRM Practices and Job Satisfaction’

1. If a researcher is interested to carry out research on topic like ‘Job Satisfaction’, then he/she can used one of the several below given questionnaires.

i. 3-items questionnaire developed by Cammann et al. (1983; attached p. 5.ii. 5-items questionnaire developed by Bacharach & Bamberger (1991;

attached page 6).iii. 7-items questionnaire developed by Cook et al. (1981; attached p. 10)iv. 6-items questionnaire developed by Pond & Geyer (1991; pp. 12-13).v. 6-items questionnaire developed by Agho et al. (1992; pp. 18-19)vi. 18-items questionnaire developed by Cook (1981; attached page 18-19).i. 5-items questionnaire developed by Rentsch & Steel (1992; p. 26)

But if researcher is interested to carry out research on topic like ‘HRM Practices and Job Satisfaction’, then he/she will have to use one of the aforementioned questionnaires along with some similarly developed questionnaires on various HRM practices.

2. Some researchers have developed mixed/hybrid questionnaires which include questions on both ‘HRM practices’ and ‘Job satisfaction’; such questionnaires are of further two categories, namely:

a. those which have mixed questions, including both aspects of job satisfaction and HRM practices, such as:ii. 20-items Minnesota Satisfaction Questionnaire (MSQ questionnaire)

developed by Weiss et al. (1967; attached pages 7-8);iii. 6-items questionnaire developed by Tsui, Egan & O’Reilly (1992;

attached page 16);iv. Job Diagnostic Survey-questionnaire developed by Hackman & Oldham

(1974; attached pages 20-22).b. those which cover questions on ‘HRM practices’ only, such as:

i. 15-items questionnaire developed by Cook et al. (1981; attached p. 27-28);ii. 36-items questionnaire developed by Spector (1997; attached p. 14-15);iii. 21-items questionnaire developed by Hatfield et al. (1985; attached p. 17).

3. The existence of the three types of questionnaire (covering questions on i. Job Satisfaction only; ii. Job satisfaction and HRM practices, and iii. HRM practices only) poses certain problems for a researcher while he/she has to select a questionnaire for adopting for research; such problems are:(a) Which questionnaire should be selected, the one having maximum number of items?

It is possible that some technically better questionnaires are available with less number of items;

(b) Should researcher combine two or more-than-two questionnaires? Then which ones? And on what basis?

47


(c) If even by combing two or more-than- two questionnaires, some particular aspects of HRM practices are still not covered, what should then researcher need to do? Econometrics theory requires all relevant variables need to be included; otherwise biased βs would be resulted.

Take-home Assignment(Due though email one day before our next class after Mid-term exam)(Hard copies of above referred pages are available at Photocopier shop)

(a) Identify questionnaires (amongst the ones referred above) which provide complete coverage of all required aspects for doing research on topic “HRM Practices and Job Satisfaction”; please also explain as to why you consider these questionnaires complete.

(b)Prepare 3-combinations of questionnaires (choosing from the above listed ones), which can provide full coverage of all aspects required on the topic. Please also explain as to why you understand that these combinations provide complete coverage of the topics or otherwise.

(c) Indicate which of the aspects of HR management (practices) are still excluded.

(d)Explain if you have some questionnaire which can provide better coverage (language-wise, contents-wise) than that of the ones referred above.

(e) In case you are supposed to do research on the above stated topic, would you like to adopt some questionnaire (which one; which combination), adapt some questionnaire (how) or develop questionnaire of your own (present a specimen).

48


Example 2

The six-dimensional Hofstede national culture: does it moderate organizational HRM practices-employees job satisfaction relationship in

Pakistani organization?

Research questions:1. Do the six-dimensions of Hofstede national culture exist in Pakistani organizatios? if yes,

then upto what extent?2. Do these cultural dimensions moderate HRM practices-employees job satisfaction

relationship in Pakistani organization?

Research objectives1. To find out the levels of prevalance of the six dimensions of Hofstede national culture in

public sector pakistani organizations.2. To check whether the prevalance of the six dimensions of Hofstede national culture

affects organizational HRM practices and employees job satisfaction in public sector pakistani organizations?

3. To identify which of the six dimensions of Hofstede national culture affects HRM practices-employees job satisfaction relationship more, relative to each others.

4. To suggest policy prescriptions based on the research findings.

Example 3

HRM and its outcomes, like:(a) HRM and employees’ commitment(b) HRM and employees’ turnover(c) Organizational justice and its outcomes lik……………(d)

49


3.3 (d) Examples in general management area

Example 1:Corporate governance practices: a cross industry comparison

(textile, pharmaceuticals, sugar and cement industries) Research questions

1. What are the general corporate governance practices in vogue in Pakistan?2. Whether such corporate governance practices influence performance in corporate sector?3. Whether corporate governance practices are industry specific? (textile, pharmaceuticals,

sugar and cement industries)

Research objectives

1. To identify various corporate governance practices in vogue in Pakistan?2. To determine the level of existence of various corporate governance practices in vogue in

Pakistan?3. To analyze the whether such corporate governance practices influence performance in

corporate sector?4. Whether corporate governance practices are industry specific? (textile, pharmaceuticals,

sugar and cement industries)

50


Topic 4

Analyzing mean values* Analyzing mean value, using one-sample t-test

* Analyzing/comparing mean-differences of two or more groups

Analyzing mean value, using one-sample t-test

Deciding whether JB variable is statistically significant?

Use SPSS command:

Analyze…comparing mean…one-sample t-test…put test-value = 3

(why?)…take JB to the right-side ‘Test-variable’ box…click OK

Paste computer output here:

One-Sample Statistics

N Mean Std. Deviation Std. Error Mean

Job satisfaction 264 4.0480 .63086 .03883

One-Sample Test

Test Value = 3

t df Sig. (2-tailed) Mean Difference

95% Confidence Interval of the

Difference

Lower Upper

Job satisfaction 26.991 263 .000 1.04798 .9715 1.1244

Interpret the results?

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COMPARING MEAN-DIFFERENCES OF TWO OR MORE

GROUPS

* TESTS for two groups and more-than-two groups are different:

* Two groups

* Independent samples t test

* Paired-sample t test

* More-than-two groups

* One-Way ANOVA

* Repeated ANOVA

* INDEPENDENT SAMPLES T TEST:

* One variable belonging to two separate samples groups,

independent of each other

* like employees job’ satisfaction across public

and private sector organizations (DO)

or across gender (DG: male = 1 & female = 0)

* INDEPENDENT SAMPLES T TEST: SPSS command is:

ANALYZE…..COMPARE MEANS…..

INDEPENDENT SAMPLE T TEST…..

Take JB to Test-variable box and DG to Group-

variable box, and define it as 1 (male) and 0

(female)….. Click Continue and OK

52


Results are:

* A pre-test for use of Independent sample t test is Levene’s test for

equality of variances, which estimates F = 2.130 at p = 0.146, suggesting

F is insignificant, so variances are equal, and Independent samples t

test can be used.

* Mean of male is 4.092, mean of female is 4.126, the mean difference

is -0.09342, and this mean difference is insignificant at t = -0.964 (p =

0.336).

53


* PAIRED –SAMPLE T TEST:

* Two variables belonging to same one group/sample

* like DJ and PJ across all respondents.

PAIRED-SAMPLE T TEST: SPSS command is:

ANALYZE…..COMPARE MEANS…..PAIRED T TEST

…..Take DJ & PJ as Variable1 and Variable2 to

Paired-Variable box…..Click OK

Results are:

* In contrast to the Independent-sample t test, wherein

equality of variances is tested using Levene’s as a pre-

test, there is no pre-test in Paired-sample t; why?

* Mean of DJ is 5.0256, mean of PJ is 4.9381, the means-

difference is 0.08878, and this means-difference is

statistically insignificant at t = 1.507 (p = 0.13).

54


COMPARING MORE-THAN-TWO GROUPS

ONE-WAY ANOVA:

* Like JB across several educational groups.

* One-way ANOVA is the extension of Independent samples t test

in case of more than two groups; in that case, SPSS’s command is:

ANALYZE…..COMPARE MEANS…..ONE-WAY

ANOVA……Take JB to Dependent and EDU to

Factor box and Click OK

* F should be significant for significant means-differences

between groups;

* POST HOC option on ONE-WAY-ANOVA , with test

Sheffe, will indicate which groups are different.

55


COMPARING MORE-THAN-TWO GROUPS

REPEATED ANOVA:

* More-than-two variables belonging to same group

* like DJ, PJ, IJ & InJ across all respondents/same one

group (whether the mean values of the four facets of

organizational justice differ across respondents)

REPEATED ANOVA T TEST: SPSS command is:

ANALYZE…..GENERAL LINEAR MODEL......

REPEATED MEASURES…..write OJ_FACETS as

Within-Subject-Factor name…..write 4 (since we

are going to test 4 facets) in Number of Levels….

click…ADD….click …DEFINE…click…..DESCRIPTIVE

STATISTICS…..Continue…..OK

Results are:

* There is a lot of stuff; important table is the

“Multivariate Tests”; all tests included here are very

significant, suggesting significant differences between

mean values of the four OJ-facets.

Take-home Assignment 4(Due in next class)

Q.1 What is the ‘one-sample t-test’ used for?Q.2 How does the use of ‘independent samples t test’ differ from that of the ‘paired-sample t

test’?Q.3 What is the Levene’s test and how is this test used?Q.4 How does the use of the test ‘One-Way ANOVA’ differ from that of ‘Repeated

ANOVA’?

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Topic 5

Uses of estimated econometric models:Some examples

(MATERIAL ON THIS TOPIC WILL BE PROVIDED

LATER-ON)

57


Topic 6

Relaxing of Standard Assumptions: Normality Assumption andits testing

In an earlier section (at the end of Topic 2), we learned about seven basic standard assumptions

of the Ordinary Least Squares (OLS) estimation technique. From this section and onwards, we

are going to learn what happens if the following four of the basic standard OLS estimation

technique are violated.

1. Normality assumption (This section

2. No multicollinearity assumption (Next

3. No heteroscadasticity assumption (three

4. No autocorrelation assumption (sections

Normality of error/disturbance term

Normality in general/normal distribution

A normal distribution, by definition, is a symmetric and bell-shaped distribution. A random

variable xi follows normal distribution, with mean equal to zero and standard deviation equal to

1. For practical purposes, the Skewness and Kurtosis of a random normal variable, respectively,

are equal to zero and 3, where the two concepts are defined, as follows.

(6.1)

where and are the estimates of third and fourth central moments, respectively, is the sample mean and is the estimate of the second central moment, the variance.

A distribution can be skewed to the left or right; if it is not skewed (S = 0), then distribution is

symmetric. Kurtosis is a measure of whether the data are peaked or flat relative to a normal

distribution. A normal distribution has Kurtosis = 3; a distribution with longer and shorter tails

relative to the normal distribution, will be having K greater than and less than 3, respectively.

58

http://en.wikipedia.org/wiki/Variance

http://en.wikipedia.org/wiki/Mean

http://en.wikipedia.org/wiki/Central_moment


Normality of error term and its tests

According to standard assumption, the error/disturbance term ei (or μi) needs to follow normal

distribution; if it does not, the use of t and F statistics, and the respective tests will not remain

valid in finite/small samples (Gujarati 2007; p. 150). However, Gujarati (2007; pp. 346-47)

further says “the usual test procedures – the t and F tests – are still valid asymptotically, that is,

in the large samples, but not in the finite or small samples”. And since researchers usually do not

have large samples, the testing of normality becomes an importance practice.

There are several ways the disturbances/residuals can be tested for normality; a few are

discussed, as follows.

i. Histogram of residuals

ii. Normal probability plot (NPP)

iii. Jarque-Bera (JB) test of normality

Histogram of residuals

It is a very simple and easy approach to visually check normality of the residuals. Let’s check the

normality of residuals using histogram of residuals of our “Organizational justice and job

satisfaction” case already introduced in section 4.2.

Let’s re-run the model:

JS = F(DJ, PJ, IJ, INJ, AEE) (6.2)

But this time we will ensure to include ‘Histogram’ in our results, using the SPSS command:

ANALYZE…..REGRESSION…..LINEAR…..(Take JS in to dependent variable box and

and DJ, PJ, IJ, INJ and AEE into independent variable box)…..PLOTS..…..

HISTOGRAM …..CONTINUE…..OK

Study the output; you will find ‘Histogram’ along with the regression results already provided in

model 4.6 (of section 4.2). Take your cursor “Histogram’, use copy command, and paste it in the

following space.

59


A visual study of the histogram reveals that the most of the residuals lie within the normal curve,

while a few residual lie outside, not only on left side, causing a little skewness, but also on top

peak, causing some Kurtosis.

60


Normal probability plot (NPP)

The following SPSS commands help draw ‘Normal probability plot’, usually abbreviated as NPP

curve.

ANALYZE…..REGRESSION…..LINEAR…..(Take JS in to dependent variable box and

and DJ, PJ, IJ, INJ and AEE into independent variable box)…..PLOTS…..NORMAL

PROBABILITY PLOT…..CONTINUE…..OK

Repeat the procedure of bringing NPP to the following place.

The interpretation of drawing NPP is that, if NPP draws in a straight line, the residuals are then

normally distributed. In the above case, the most part of the NPP (which is also referred to as

Normal P-P Plot in econometric literature) seems to be approximately in a straight line, with the

exception of a small part which does not coincide exactly with the straight line.

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Jarque - Bera Normality test

Jarque and Bera (1987)12 made use of the aforementioned Skewness and Kurtosis concepts and

developed the famous Jarque–Bera test for testing the normality of disturbance term; their test

statistic JB is defined, as:

where n is the number of observations (or degrees of freedom in general); S is the sample

Skewness, and K is the sample Kurtosis.

The JB statistic asymptotically follows chi-squared distribution, with degrees of freedom = 2.

However, it should be noted that the JB test is an asymptotic or large sample sized test; it may

not work in smaller samples.

One can measure JB after calculating S and K; a number of good econometric software include

JB test in their routine regression tests.

12 Jarque, C.M. and Bera, A.K. (1987). “A Test for Normality of Observations and Regressions Residuals, International Statistical Reviews, 55:163-172

62

http://en.wikipedia.org/wiki/Degrees_of_freedom_(statistics)

http://en.wikipedia.org/wiki/Chi-squared_distribution

http://en.wikipedia.org/wiki/Asymptotic_analysis

http://en.wikipedia.org/wiki/Kurtosis

http://en.wikipedia.org/wiki/Skewness

http://en.wikipedia.org/wiki/Test_statistic

http://en.wikipedia.org/wiki/Test_statistic


Outliers: exploring the data

What is an outlier?

In the language of Gujarati (2007; p. 399), “an outlying observation, or outlier, is an observation

that is much different (either very small or very large) in relation to the observations in the

sample. More precisely, an outlier is an observation from a different population to that

generating the remaining sample observations. The inclusion or exclusion of such an

observation, especially if the sample size is small, can substantially alter the results of regression

analysis”.

The following SPSS commands can help us to identify certain outlying observations in our data

set.

ANALYZE.....DESCRIPTIVE STATISTICS.....EXPLORE......(Take JB13 to right-hand

‘Dependent List’ box and go to Statistics).....STATISTIC.....Click on

OUTLIER......CONTINUE......PLOT.....Cllick on Stem & Leaf, Histogram and Normalty

Plot with test.......CONTINUE.....(on-display, pick).....BOTH....OK.

13 In contrast to the earlier cases of Histogram, NPP and JB test wherein we were interested to check the normality of residuals obtained from regressing JB over DJ, PJ, IJ and INJ, we are now directly checking the outlying observations in only one - the dependent variable (JB).

63


The above noted SPSS commands give us a lot of information/materials, including the following:

1. Table entitled DESCRIPTIVES:

Descriptives

Statistic Std. Error

Job satisfaction Mean 4.0480 .03883

95% Confidence Interval for Mean

Lower Bound 3.9715

Upper Bound 4.1244

5% Trimmed Mean 4.1028

Median 4.1667

Variance .398

Std. Deviation .63086

Minimum 1.17

Maximum 5.00

Range 3.83

Interquartile Range .67

Skewness -1.592 .150

Kurtosis 4.224 .299

The mean value of the employees’ responses on job satisfaction averages at 4.048; the vale falls

between 4 (I Agree) and 5 (I strongly Agree). The values of Skewness (S) and Kurtosis (K),

respectively are -1.592 and 4.224, while a normal distribution requires these values to be equal to

0 and 3.

2. A table with EXTREME VALUES:Extreme Values

Case Number Value

Job satisfaction Highest 1 11 5.00

2 55 5.00

3 88 5.00

4 150 5.00

5 184 5.00a

Lowest 1 229 1.17

2 31 1.17

3 228 1.50

4 198 2.00

5 196 2.17

a. Only a partial list of cases with the value 5.00 are shown in the table of upper extremes.

The highest extreme values in this case are logically acceptable, but the value of observation No. 31 and 229 are extremely low, each one is equal to 1.17; a third observation No.228 als has a low value (1.50.

64


3. Results of the normality tests, namely:

Tests of Normality

Kolmogorov-Smirnova Shapiro-Wilk

Statistic df Sig. Statistic df Sig.

Job satisfaction .155 264 .000 .880 264 .000

a. Lilliefors Significance Correction

Out of the two tests, the latter test (Shapiro-Wilk Test) is considered more appropriate for small sample sizes (< 50 samples) but it can also handle sample sizes as large as 2000.

In both test cases, if the Sig. value of is greater than 0.05, then the data is normal. If it is below 0.05, then the data significantly deviate from a normal distribution, as is in our case.

65


4. Histrogram

It reflects that most of the responses lie within the values of 3 and 5, with the exception

of a few which appear lying on extreme left side, between values of 1 and 2.

66


5. Stem and Leaf Plot:

Job satisfaction Stem-and-Leaf Plot

Frequency Stem & Leaf

16.00 Extremes (=<2.8) 4.00 3 . 0011 8.00 3 . 33333333 8.00 3 . 5555555522.00 3 . 666666666666666666666625.00 3 . 888888888888888888888888875.00 4 . 000000000000000000000000000000000000000001111111111111111111111111111111137.00 4 . 333333333333333333333333333333333333325.00 4 . 555555555555555555555555521.00 4 . 66666666666666666666612.00 4 . 88888888888811.00 5 . 00000000000

Stem width: 1.00Each leaf: 1 case(s)

This plot reinforces that there are some extreme cases especially on lower side,

suggesting that 16 percent responses came with the value of below 3.

67


Normal Q.Q. Plot

In order to determine normality graphically we can use the output of a normal Q-Q

Plot. If the data are normally distributed then the data points will be close to the

diagonal line. If the data points stray from the line in an obvious non-linear fashion

then the data are not normally distributed. From this graph we can conclude that the

data mostly appear to be normally distributed as it follows the diagonal line with the

exception of some portions where data appear away from the straight diagonal line.

The detrended Normal Q-Q Plot, provided below, further clarifies the position.

68


7. Detrended Normal Q.Q Plot

69


6. Box

The box plot discriminates between majority of the cases which lied between values of 3

to 5, and ones fell below 3; this plot helps identify all the cases having values below 3, as

well as, the three cases having values below 2.

Take-home Assignment 6Repeat the exercise after dropping the three extreme cases (31, 228

& 229), and note whether some improvement occurred.

70


Topics 7 - 9

MULTICOLLINEARITY, HETROSCADASTICITY AND AUTOCOLLINERAITY: THREE MAJOR

ECONOMETRICS PROBLEMS, THEIR NATURE, DETECTION AND REMEDIES

71


Topic 7

Evaluating estimated model using econometrics criteriaProblem of multicollinearity: what happens if

regressors are correlated?

Multicollinearity: what is it?

According to one of the standard assumptions of the Ordinary Least Squares (OLS) estimation

technique already discussed in Topic 2, the explanatory variables, X i should not linearly correlate

or affect each others; if they do, the problem is referred to as multicolinearity problem. In

regression, we assume:

Y = β0 + β1X1 + β2X2 + β3X3 ………e (7.1)

That is, Y depends on X1, X2, X3 ………; but in case of the existence of multicollinearity, two

or more explanatory variables do correlate, like:

X1 = β0 + β2X2 + β3X3 + β4X4 ……… (7.2)

That is, X1 depends on X2, X3, …… and respective β2, β3 … are found statistically significant,

and/or

X2 = β0 + β1X1 + β3X3 + β4X4 ……… (7.3)

That is, X2 depends on X1, X3, …… and respective β1, β3 … are turned out to be statistically

significant.

Multicollinearity is thus not a problem originated from or related to the specification of the

model or the estimation of the specified model, it is a problem originating from the nature of the

data as it exists/happens in case when one (or more) explanatory variable affects other

explanatory variable(s). In practice, one can reduce multicollinearity, he/she cannot altogether

eliminate it.

We should therefore be interested in knowing the fact whether multicollinearity perfectly exists

or less than perfectly. In case, the explanatory variables are perfectly collinear, the regression

coefficients will be indeterminate, as their standard errors are infinite. In case, multicolinearity is

72


less than perfect, the regression coefficients, although indeterminate, will possess large standard

errors, meaning the coefficients cannot be estimated with great precision or accuracy.

Let’s try to understand the nature of the perfect collinear and less-than-perfect collinear

explanatory variables. Table 7.1 provides data on Y and three intended explanatory variables,

namely X1, X2, X3 and X4.

Table 7.1Y X1 X2 X3 X4

1100 10 30 50 571250 15 45 75 791376 18 54 90 1111574 24 72 120 1311895 30 90 150 143

Note that we have X2 and X3 multiple of X1, respectively, by 3 and 5 times, so these three are

perfectly correlated and X4 is not; estimating the correlation, using the following commands:

ANALYZE…..CORRELATE…..BIVARIATE…..(take X1, X2, X3 and X4 to the right

side of the box)…..click OK; study the output.

Correlations

X1 X2 X3 X4X1 Pearson Correlation 1 1.000(**) 1.000(**) .966(**) Sig. (2-tailed) .000 .000 .007 N 5 5 5 5X2 Pearson Correlation 1.000(**) 1 1.000(**) .966(**) Sig. (2-tailed) .000 .000 .007 N 5 5 5 5X3 Pearson Correlation 1.000(**) 1.000(**) 1 .966(**) Sig. (2-tailed) .000 .000 .007 N 5 5 5 5X4 Pearson Correlation .966(**) .966(**) .966(**) 1 Sig. (2-tailed) .007 .007 .007 N 5 5 5 5

** Correlation is significant at the 0.01 level (2-tailed).

The output reflects 100 percent correlation between the first three Xs, and a little lesser between

X4 and the first three Xs.

Let’s regress Y on the four explanatory variables, using SPSS command:

73


ANALYZE…..REGRESSION…..LINEAR…..(take Y into dependent variable box and

X1, X2, X3 and X4 into the independent variable box)…..click OK.

Check what happens: regression process takes which of the explanatory variables in to its

estimation and which not.

Consequences of multicollinearity

1. Although BLUE, the OLS estimators have large variances and covariances, making

precision estimation difficult.

2. Because of the aforementioned consequence, the confidence intervals tend to be much

wider, leading to the acceptance of zero null hypothesis more readily.

3. The t ratios of one or more coefficients tend to be statistically insignificant.

4. R2 is very high.

5. The OLS estimators (βs), t ratios and their standard errors are sensitive to small changes.

Detection of multicollinearity

As already mentioned, Multicollinearity is not a problem relating to the specification of model or

its estimation; it is a problem originating from the nature of the data as it exists/happens when

one X affects another X. In practice, one cannot altogether eliminate multicollinaearity, so its

detection should mean to locate which one or two explanatory variables are causing the problem,

and what the degree or level of collinearity exists between such variables. Such detection of the

problem may help reduce the severity of the problem.

There are a number of measures which can be used to measure the level or degree of

multicollinearity; we however discuss the following ones.

1. Rule of thumb: High R2 and insignificant t-ratios

2. Correlation between X-variables

3. Auxilliary regressions

4. Klien”s rule of thumb: multicollinearity is troublesome only if R2 from auxiliary-

regression > R2 from regular-regression

5. Tolerance and VIF

6. Eigenvalues and CI

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Rule of thumb: High R2 and insignificant t-ratios

When R2 is reasonably high and F-statistic significant, but a large number of individual

coefficients βi are statistically insignificant, this phenomenon reflects the existence of the

problem of multicollinearity.

Using correlation between X-variables

Estimating correlation between explanatory variable of ‘Organizational justice and job

satisfaction’:

Correlations

Distributive justice

Procedural justice

Interactive justice INJ AEE

Distributive justice

Pearson Correlation 1 .684** .505** .571** .206**

Sig. (2-tailed) .000 .000 .000 .001

N 264 264 264 264 264

Procedural justice

Pearson Correlation .684** 1 .564** .660** .134*

Sig. (2-tailed) .000 .000 .000 .029

N 264 264 264 264 264

Interactive justice

Pearson Correlation .505** .564** 1 .543** .111

Sig. (2-tailed) .000 .000 .000 .071

N 264 264 264 264 264

INJ Pearson Correlation .571** .660** .543** 1 .122*

Sig. (2-tailed) .000 .000 .000 .047

N 264 264 264 264 264

AEE Pearson Correlation .206** .134* .111 .122* 1

Sig. (2-tailed) .001 .029 .071 .047

N 264 264 264 264 264

**. Correlation is significant at the 0.01 level (2-tailed).

*. Correlation is significant at the 0.05 level (2-tailed).

Auxilliary regression:

Since multicollinearity arises because one or more of the regressors are exact or approximately

linear combinations of other regressors, each of the regressors is regressed on all other

regressors, R2 of each of the auxiliary regressions is obtained and respective F-statistics are

calculated, using the following formulas.

Fi = {R2/(k-2)}/{(1-R2)/(n-k+1)} (7.4)

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If respective F statistic, calculated using formula (7.4), is found significant (calculated F i >

Ftabulated), the respective X variable is considered correlated with other explanatory variables,

causing problem of multicollinearity (Gujarati 2007; p369).

Let’s run auxiliary regressions of the “Organizational justice and job satisfaction” case already

introduced in section 4.2; the original model is:

JS = F(DJ, PJ, IJ, INJ, AEE) (7.5)

Since there are five explanatory variables, we would have to run five auxiliary regressions,

namely:

DJ = F(PJ, IJ, INJ, AEE) (7.6a)

PJ = F(DJ, IJ, INJ, AEE) (7.6b)

IJ = F(DJ, PJ, INJ, AEE) (7.6c)

INJ = F(DJ, PJ, IJ, AEE) (7.6d)

AEE = F(DJ, PJ, IJ, INJ) (7.6e)

Running regressions 7.6 (a – e) would yield the following R2:

R2DJ = 0.516 (7.7a)

R2PJ = 0.596 (7.7b)

R2IJ = 0.383 (7.7c)

R2INJ = 0.494 (7.7d)

R2AEE = 0.040 (7.7e)

Calculating respective F, using the formuala already given in (7.4):

FDJ ={R2/(k-2)}/{(1-R2)/(n-k+1)} (7.8a)

= {0.516/(4-2)}/{(1-0.516)/(264-4+1) (7.8b)

= {0.516/2}/{(0.484)/(261) (7.8c)

= {0.258}/(0.001854) (7.8d)

= 139.1281 (7.8e)

F-calculated = 139.1281 > F-tabulated = 4.61, with DF = 2 & 261 at p < 0.01, suggesting explanatory

variable DJ is strongly correlated with other explanatory variables.

FPJ ={R2/(k-2)}/{(1-R2)/(n-k+1)} (7.9a)

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= {0.596/(2)}/{(0.404)/(261)} (7.9b)

= {0.298}/{(0.001548)} (7.8c)

= 192.5198 (7.8e)


variable PJ is strongly correlated with other explanatory variables.

FIJ ={R2/(k-2)}/{(1-R2)/(n-k+1)} (7.10a)

= {0.383/(2)}/{(0.617)/(261)} (7.10b)

= {0.1915}/{(0.002364) (7.10c)

= 81.00729 (7.10e)


variable IJ is strongly correlated with other explanatory variables.

FINJ ={R2/(k-2)}/{(1-R2)/(n-k+1)} (7.11a)

= {0.494/(2)}/{(0.506)/(261)} (7.11b)

= {0.247}/{(0.001939) (7.11c)

= 127.4051 (7.11e)


variable INJ is strongly correlated with other explanatory variables.

FINJ ={R2/(k-2)}/{(1-R2)/(n-k+1)} (7.12a)

= {0.040/(2)}/{(0.960)/(261)} (7.12b)

= {0.020}/{(0.003678) (7.12c)

= 5.4375 (7.12e)


variable INJ is moderately correlated with other explanatory variables.

Klien’s rule of thumb

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According to Klien (1962)14, multicollinearity is troublesome only if R2from auxiliary-regression

is greater than the R2 obtained from the regular regression of Y on Xs.

We have calculated R2 from our five auxiliary regressions in our previous section; these are:

R2DJ = 0.516 (7.13a)

R2PJ = 0.596 (7.13b)

R2IJ = 0.383 (7.13c)

R2INJ = 0.494 (7.13d)

R2AEE = 0.040 (7.13e)

We have also already calculated our regular main regression’s R2 equal to 0.2560 in our previous

section 4.2. With the exception of one auxiliary regression R2AEE = 0.040, all other auxiliary

regression R2s have been found greater than the regular one.

Tolerance and VIF

The word ‘TOLERANCE’ means broadmindedness, open-mindedness, patience or ‘to tolerate’.

In econometrics, TOLERANCE, or its abbreviation, TOL has special use, and is measured as:

TOL = 1 – R2J (7.14)

where R2J is R2 obtained in auxiliary regressions, the regressions wherein one explanatory

variable is regressed over other explanatory variables (Gujarati, 2007; pp.358-371).

In case of perfect collinearity amongst two explanatory variables R2J will measure equal to 1, and

TOL = 0; and in case of zero-collinearity, R2J will measure equal to 0, and TOL = 1;

summarizing:

In case of perfect-collinearity (R2J = 1): TOL = 1 – R2

J = 0 (7.15)

In case of zero-collinearity (R2J = 0): TOL = 1 – R2

J = 1 (7.16)

Hence in case of imperfect-collinearity (0 < R2J < 1),

TOL will increase as far as R2J decreases (and vice versa) (7.17).

TOL has an inverse relationship with ‘variance-inflating-factor’, abbreviated as VIF, like:

VIF = 1 / TOL or TOL = 1 / VIF (7.18)

The SPSS’s regression output can provide statistics on TOL and VIF, if regression is run with an

additional option ‘COLLINERITY DIAGNOSTICS’ in statistics.

14 Klien, L.R. (1962). An Introduction to Econometrics. Prentice-Hall, Englewood Cliffs, N.J. p.101; also reported in Gujarati, (2007; p.369).

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The results of ‘Collinearity statistics (TOL & VIF)’ should be interpreted, using the following

rules of thumb.

1. TOL ranges between 0 and 1, that is: 0 < TOL < 1; hence:

a. The closer is TOL to zero, the greater is the degree of collinearity of that explanatory

variable with other explanatory variables; hence, we can identify which one of the

explanatory variables is contributing the highest collinearity.

b. The closer is TOL to 1, the greater is the evidence of non-collinearity of that

explanatory variable with other explanatory variables.

2. TOL and VIF are inverse to each other, that is:

VIF = 1 / TOL = 1 / (1 – R2J) (7.19)

a. If R2J = 0 (zero-collinearity), then TOL = 1, and VIF = 1 (so VIF has the lowest level

= 1).

If R2J = 1 (perfect collinearity), then TOL = 0, and VIF = ∞ (VIF goes to infinity).

So VIF ranges between 1 and ∞.

b. If R2J = 0.00 TOL = 1 - R2

J = 1 & VIF = 1 / TOL = 1

If R2J = 0.25 à TOL = 0.75 & VIF = 1.33

If R2J = 0.50 à TOL = 0.50 & VIF = 2.00

If R2J = 0.75 à TOL = 0.25 & VIF = 4.00

If R2J = 0.90 à TOL = 0.10 & VIF = 10.00

If R2J = 0.95 à TOL = 0.05 & VIF = 20.00

If R2J = 0.99 à TOL = 0.01 & VIF = 100.00

If R2J = 1.00 à TOL = 0.00 & VIF = ∞ (7.20)

It appears from the above analysis that, whereas auxiliary regression’s coefficient of

determination R2J and its resultant TOL have inverse relationship (the former

increases from zero to 1, the latter decreases from 1 to 0), the relationship between R2J

and VIF is positive and direct (the former increases from 0 to 1, the latter increases

from 1 to ∞).

c. It is worth-noting that value of VIF substantially increases with an increasing rate, at

each point of increase in R2J; so multicollinearity would become a more troublesome

problem at higher levels of R2J..

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Let’s rerun our “Organizational Justice and Employees’ Job Satisfaction’ case, and check it for

the problem of multicollinearity, using the TOL and VIF statistics discussed as above.

Eigenvalues and CI

The SPSS’s ‘Collinerity Diagnostics’ command, already referred to, also provides statistic on

‘Eigenvalues’ and ‘Condition Index (CI)’. CI is derived on the basis of Eigenvalues. According

to Gujarat (2007; pp.369-70), the rule of thumb for the use of CI is:

a. There would be moderate to strong multicollinearity if CI falls within a range of 10 to

30.

b. Multicollinearity would be severe if CI exceeds 30.

Check whether the data used for the case of “Organizational Justice and Employees’ Job

Satisfaction’ suffer from the problem of multicollinearity.

Take-home assignment 7Study section 10.8 on ‘Remedial Measures’ by Gujarati (2007; pp.371-77) and prepare your own notes on the topic: ‘Remedial Measures of Multicollinearity Problem: Important Points’; submit

a copy as next take-home assignment.

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Topic 8Evaluating Estimated Model Using Econometrics Criteria

Problem of Heteroscadasticity: What Happens ifThe Error Variance is Nonconstant?

Nature of the Problem:

Like no-muticolinearity assumption, no-heteroscadasticty is another important

assumption of the classical linear estimation technique. This assumption is also referred

to as the assumption is homoscedasticity, where ‘homo’ means equal and ‘scedasticity’

means spread or variance. Homoscedasticity thus refers to as equal or same variances.

===> E(ui²) = σ²; σ² remains constant while σ²i varies

In case, σ² is not constant, we face a problem referred to as “Heteroscedasticity”.

There are several reasons why the variances of are variable: some of these reasons are, as

follows:

a) As people learn and become experts, their error of behavior become smaller

overtime. In this case, variances are expected to decrease.

b) As income grows, people have more choices about the disposition of their

incomes. Hence variances are likely to increase with increase in income.

c) As data collecting techniques improve, variances are expected to decrease.

It should be noted that the problem of heteroscedasticity is likely to be more common in

cross-sectional than time-series data. In cross-sectional data, one collects data at a given

point in time, and the data are collected from respondents who generally differ in several

respects.

Consequences of heteroscadasticity:

1) Due to non-constant or variable nature of the variance, variances of ßi are larger,

and consequently, their standard errors and confidence interval are large, while t

ratios are consequently small and insignificant.

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2) Estimated results are misleading.

3) OLS estimators are no longer efficient, not even asymptotically.

Detection of heteroscadasticity:

Nature of the problem:

In cross- sectional data, where we have to collect data on micro, small, medium and large

farms/firms, heteroscedasticity is likely to be there.

Park Test:

Run a usual regression, like:

lnY = ß0 + ß1lnXi + μi (8.1)

Obtain residuals ei and make them squared, run regression of the following form:

Lne2i = ß0 + ß1lnXi + μi (8.2)

If ß1 happens to be statistically significant, it will indicate the existence of the problems

of heteroscedasticity. Let’s do the Park test for evaluating our ‘Job satisfaction and

organizational justice’ case for checking existence of heteroscadasticity problem.

Convert data on all dependent and independent variables JB, DJ,PJ, IJ, INJ and AEE into

log using TRANSFORM and COMPUTE VARIABLE commands in SPSS; let the newly

log-variables have new names LJB, LDJ,LPJ, LIJ, LINJ and LAEE.

Regressing (8.1) type of model:

lnLB = ß0 + ß1lnDJ + ß2lnPJ + ß3lnIJ + ß4lnIN + ß5lnAEE + μi (8.3)

Obtain residuals using additional SPSS commands: ANALYZE…REGRESSION …

LINEAR…SAVE…RESIDUALS…UNSTANDARDIZED…CONTINUE…OK

This command will estimate residuals and put those in the last column of the data file

under name ‘RES_1’. Make this variable square (as we need Lne2i as per equation 8.2),

using TRANSFORM and COMPUTE commands.

Now you can run regression on the second equation, like (8.2); doing so:

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Lne2i = ß0 + ß1lnDJ + ß2lnPJ + ß3lnIJ + ß4lnIN + ß5lnAEE + μi (8.4)

We get results like:

Coefficientsa

Model

Unstandardized CoefficientsStandardized Coefficients

t Sig.B Std. Error Beta

1 (Constant) .240 .098 2.450 .015

LDJ -.157 .026 -.455 -6.124 .000

LPJ -.008 .022 -.027 -.341 .733

LIJ .026 .024 .069 1.075 .283

LINJ -.056 .032 -.129 -1.748 .082

LAEE .021 .024 .046 .848 .397

a. Dependent Variable: Lnes

The three coefficients (LPJ, LIJ & LAEE) are statistically insignificant while two coefficients

(LDJ & LINJ) are statistically significant, suggesting the possibility of moderate level of

heteroscadasticity problem.

Goldfeld-Quant Test:

The Goldfeld-Quant test suggests ordering or rank observations according to the values

of Xi, beginning with the lowest Xi value. Then some central observations are omitted in

a way that the remaining observations are divided into two equal groups. These two data

groups are used for running two separate regressions, and residual sum of squares (RSS)

are obtained; these RSSs (RSS1 & RSS2) are then used to compute Goldfeld-Quant F test,

namely:

(8.5)

If the F is found significant (F-calculated > F-tabulated, the problem of heteroscedasticity

is likely to exist.

Let’s run the stated test for ‘Organizational justice and Job satisfaction’ case. The

aforementioned Park’s test indicated that log of variable DJ was found the most collinear

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with the log of the squared residuals; this suggested that we arrange data in ascending

order using DJ variable as the base, and then omit central 14 observations, which will

leave 250 observation to be equally divided in two parts of 150 observation each.

The SPSS command is: DATA…SORT CASES…Take DJ to the ‘SORT-BY’ BOX…

ASCENDING.

Remove the 14 central observations, and save data in two separate files, one having

Group 1 data (the first 150 observations) and the second having Group II data (having

150 later observations).

Then running the required two regressions gives the following TWO ANOVA tables:

GROUP – I: ANOVAb

Model Sum of Squares Df Mean Square F Sig.

1 Regression 14.897 5 2.979 6.447 .000a

Residual 54.995 119 .462

Total 69.892 124

a. Predictors: (Constant), AEE, Procedural justice, Interactive justice , Distributive justice, INJ

b. Dependent Variable: Job satisfaction

GROUP – II: ANOVAb

Model Sum of Squares Df Mean Square F Sig.

1 Regression 4.123 5 .825 5.005 .000a

Residual 19.605 119 .165

Total 23.728 124

a. Predictors: (Constant), AEE, Distributive justice, Interactive justice , INJ, Procedural justice

b. Dependent Variable: Job satisfaction

The residual sum of squares (RSS) of the two groups are:

RSS1 = 54.995 with DF = 119

RSSII = 19.605 with DF = 119

Calculating F, using (8.5) F = (RSSII/DF) / (RSSI/DF)

= (19.605/119) / 54.995/119

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= 0.3565 (8.6)

F-calculated = 0.3565 < F-tabulated = 1.29 (at p = 0.05), suggesting there exists no

heteroscadasticity.

White’s General Heteroscedasticity Test

Unlike the Goldfeld–Quandt test, which requires reordering the observations with respect to the

X variable that supposedly caused heteroscedasticity, or the BPG test, which is sensitive to the

normality assumption, the general test of heteroscedasticity proposed by White does not rely on

the normality assumption and is easy to implement. As an illustration of the basic idea, consider

the following three-variable regression model.

Yi = β1 + β2X2i + β3X3i + ui (8.7)

Step 1: Given the data, we estimate (8.7) and obtain the residuals, ui.

Step 2: We then run the following (auxiliary) regression:

u2i = α1 + α2X2i + α3X3i + α4X2

2i + α5X23i + α6X2iX3i + vi (8.8)

Obtain the R2 from this (auxiliary) regression.

Step 3: Under the null hypothesis that there is no heteroscedasticity, thatis:

n R2 ~ asy χ2df (8.9)

where df is the number of regressors (excluding the constant term) in the auxiliary regression. In

our example, there are 5 df since there are 5 regressors in the auxiliary regression.

Step 4. If the chi-square value obtained in (8.9) exceeds the critical chi-square value at the

chosen level of significance, the conclusion is that there is heteroscedasticity. If it does not

exceed the critical chi-square value, there is no heteroscedasticity.

Gujarati (2007, pp.422) advises caution in using the White test; he says: the White test can be a

test of (pure) heteroscedasticity or specification error or both. It has been argued that if no cross-

product terms are present in the White test procedure, then it is a test of pure heteroscedasticity.

If cross-product terms are present, then it is a test of both heteroscedasticity and specification

bias.

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Remedies:

1) If we know σ², then we use the weighted least squares (WLS) estimation

technique, i.e.,

(8.7)

Where σi = standard deviation of the Xi.

2) Log -transformation:

(8.8)

It reduces the heteroscedasticity.

3) Other transformations:

a) (8.9)

After estimating the above model, both the sides are then multiplied by Xi.

b) (8.10)

Note: In case of transformed data, the diagnostic statistics t- ratio and F- statistic

are valid only in large sample size.

Take-home Assignment 8

Apply the solutions provided in (8.7) to (8.10), and comment on the improvements made,

if any.

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Topic 9Evaluating Estimated Model Using Econometrics Criteria

Problem of Autocorrelation: What Happens ifthe Error Terms are Correlated?

Autocorrelation?

In accordance with one of the major assumptions of classical regression model, the ‘error term’

of one observation should be independent of the error term of other observation, i.e., μi and μj

should not correlate; mathematically:

Cov(μi and μj) = 0 (9.1)

This is no-serial-autocorrelation assumption. However, when this assumption is violated and the

two error terms are correlated, then we face the problem of autocorrelation. If such a correlation

is observed in cross-sectional data, it is called spatial autocorrelation, but spatial autocorrelation

occurs by chance, not usually. It is the time series data where chances of the occurrences of

autocorrelation are great.

In case, error terms are plotted against time (Gujarati, 2007; Figure 12.1, page 454):

μ μ + μ

+ + + + + + + + + + + + + time + time + time + + + + + + + + + + + + Panel (a) Panel (b) Panel (c)

μ μ + + + + + + ++ + + + time + + +++ + + + time + + + + + + + +

Panel (d) Panel (e)

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Panels a – d show specific patterns; panel (a) shows a cyclic pattern, panels (b) and (c) show an

upward and downward linear trend, and pane (d) indicates both linear and quadratic trend

patterns. All these cases indicate specific pattern of error terms and possibility of occurrence of

the autocorrelation problem. Against all such cases, panel (e) does not show any systematic

pattern, indicating no autocorrelation.

Consequences

1. The residual variance is likely to underestimate the true variance σ2.

2. As a result, we are likely to overestimate R2.

3. Var(βi) underestimates.

4. Consequently, t and F tests are no longer valid; these mislead about the statistical

significance of estimated regression coefficients.

An Example: In case, we want to know the relationship between real compensation (Y)

and productivity (X), using the data provided in Table 12.4 (Gujarati 2007, p. 470).

Y X58.5 47.259.9 48.061.7 49.863.9 52.165.3 54.167.8 56.669.3 58.671.8 61.073.7 62.376.5 64.577.6 64.879.0 66.280.5 68.882.9 71.084.7 73.183.7 72.284.5 74.887.0 77.288.1 78.489.7 79.590.0 79.789.7 79.8

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89.8 81.4 91.1 81.2 91.2 84.0 91.5 86.4 92.8 88.1 95.9 90.7 96.3 91.3 97.3 92.4 95.8 93.3 96.4 94.5 97.4 95.9100.0 100.0 99.9 100.1 99.7 101.4 99.1 102.2 99.6 105.2101.1 107.5105.1 110.5

Y = f(X) = β0 + β1X + e (9.2)

Estimating (9.2),

Model Summary(b)

Model R R Square

Adjusted R Square

Std. Error of the

EstimateDurbin-Watson

1 .979(a) .958 .957 2.67553 .123

ANOVA(Model

Sum of Squares Df

Mean Square F Sig.

1 Regression 6274.757 1 6274.757 876.549 .000(a) Residual 272.022 38 7.158 Total 6546.779 39

CoefficientsΒ SE t Sig

Constant 29.519 1.942 15.198 0.000

X 0.714 0.024 29.607 0.000

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Model is statistically significant (F = 876.549; p , 0.01); R2 is very good; t statistic is very

significant (p , 0.01); however, DW = 0.123, indicating that the model is mis-specified or is

suffering from autocorrelation problem.

Checking for mis-specification

There are several ways for checking of mis-specification of a model; we apply the following

three methods:

(a) Trying in Log-linear form

lnY = β0 + β1lnX + e (9.3)

Estimating model (9.3):

Model Summary(b)

Model R R Square

Adjusted R Square

Std. Error of the

EstimateDurbin-Watson

1 .987(a) .975 .974 .02605 .154

ANOVAModel

Sum of Squares Df

Mean Square F Sig.

1 Regression .995 1 .995 1466.062 .000(a) Residual .026 38 .001 Total 1.021 39


Constant1.524 .076 19.995 .000

lnX.672 .018 38.289 .000

The model relatively improved in terms of F statistic and t ratio, but DW statistic remains

suggesting the existence of the problem.

(b) Incorporate trend (t)

Y = β0 + β1X + β2t + e (9.4)

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Model Summary(b)

Model R R SquareAdjusted R

SquareStd. Error of the Estimate Durbin-Watson

1 .981(a) .963 .961 2.55661 .205

ANOVA

Model Sum of

Squares Df Mean Square F Sig.1 Regression 6304.938 2 3152.469 482.305 .000(a) Residual 241.841 37 6.536 Total 6546.779 39


Constant 1.475 13.182 0.112 0.912

X

T

1.306

-0.903

0.276

0.420

4.723

-2.149

0.000

0.038

The results have improved; trend t has been turned out statistically significant; but DW = 0.205

is still suggesting same problem.

(c) Using X-variable in quadratic form

Y = β0 + β1X + β2X2 + e (9.5)


Model Summary

Model R R SquareAdjusted R

SquareStd. Error of the Estimate Durbin-Watson

1 .997(a) .995 .994 .96689 1.030

ANOVA

Model Sum of

Squares df Mean Square F Sig.1 Regression 6512.188 2 3256.094 3482.880 .000(a)

Residual 34.591 37 .935

Total 6546.779 39

Coefficients

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Β SE t Sig

Constant -16.2182.955 -5.489 0.000

XX2

1.949-0.008

0.0780.000

24.987-15.936

0.0000.000

Specification of the model has improved; but DW statistic is still indicating problem.

In all the three cases, DW is very low relative to the desired value of DW = 2 (or near to 2);

hence, there seems existence of autocorrelation problem relative to the specification one. There

are a number of methods and tests used for detection of autocorrelation; let’s try a few such

tools/tests.

Detecting Autocorrelation

1. Plotting residuals

Using the following SPSS command, we can estimate and save the residual of regression

analysis in our data file.

ANALYZE…REGRESSION LINEAR…SAVE…RESIDUALS …

UNSTANDARDIZED…CONTINUE…OK

A visual study of the residuals (in data table), as well as, their plotting against the actual time or

trend (T), like the following one, indicates existence of a set pattern in residuals, which suggests

problem of autocorrelation.

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(2) The Runs test

The runs or Geary test is a non-parametric test used to detect autocorrelation problem. We have

already saved regression residuals. We now use the following SPSS command to run the runs

test.

ANALYZE…NONPARAMETRIC TESTS…take saved residuals to test-variable list

box…click MEAN…OK

The output box shows:Runs Test

Unstandardized Residual

Test Valuea .0000000

Cases < Test Value 19

Cases >= Test Value 21

Total Cases 40

Number of Runs 3

Z -5.605

Asymp. Sig. (2-tailed) .000

a. Mean

The output box indicates that:a. There are 19 negative sign cases (out of total

b. There are 21 positive sign cases (40 cases

c. Number of runs are = 3

The number of runs should lie between Z = ± 1.96 for no-autocorrelation; our Z = - 5.605

indicates the mean-runs are lying outside the critical region; hence results suggest existence of

the problem of autocorrelation.

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(3) Using DW statistic

The Durban-Watson d or DW statistic ranges between 0 and 4; where:

a. There is no-autocorrelation around a d = 2 (between du and 4-du)

b. Then there are two ‘indecision zones’ on both sides of ‘No-autocorrelation’ zone.

c. On both extreme ends, ‘positive autocorrelation’ and ‘negative autocorrelation’ zones

exist.

[ ] [ ] + [ Indecision ] No [ Indecisive ] - Autocorrelation [ Zone ] Autocorrelation [ Zone ] Autocorrelation

[ ] [ ]0 __________dl__________du________2______4-du_________4-dl____________ 4

How to test? The estimated model (9.2) estimates DW = 0.123, which needs to compare with

the tabulated values provided in the Durban-Watson d statistic tables. We have n = 40 and K’ = 1

(k excluding intercept). At n = 40 and K’= 1, table provides dl = 1.442 and du = 1.544. As

calculated DW = 0.123 falls below du, that suggests existence of the problem of autocorrelation.

Remedies (Gujarati 2007, pages 485-495)

There are two major remedies, namely:

(a) When the ‘coefficient of autocorrelation’ (rho = ρ) is not known, then remedy is

‘first-differencing’, that is:

(Yt – Yt-1) = β1(Xt – Xt-1) + et (9.6a)

(b) When ρ is known, then remedy is:

(Yt – ρYt-1) = α + β1(Xt – ρXt-1) + et (9.6b)

The First-Differencing method

Using TRANSFORM and COMPUTE command in SPSS, we can generate lagged variables,

namely:

LagY = Yt-1

LagX = Xt-1

Further generating FDY = Yt – Yt-1 = Yt – LagY (9.7a)

and FDX = X = Xt-1 = Xt – LagX (9.7b)

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Running regression:

FDY = α + β1FDX + et (9.8)

Results are:

Model Summaryc,d

Model R R Squareb

Adjusted R

Square

Std. Error of the

Estimate Durbin-Watson

1 .831a .690 .683 .92580 1.611

a. Predictors: FDX

b. For regression through the origin (the no-intercept model), R Square measures the

proportion of the variability in the dependent variable about the origin explained by

regression. This CANNOT be compared to R Square for models which include an intercept.

c. Dependent Variable: FDY

d. Linear Regression through the Origin

Residuals Statisticsa,b

Minimum Maximum Mean Std. Deviation N

Predicted Value -2.9518 .6480 -1.1393 .76208 40

Residual -1.84013 2.14796 -.02567 .92543 40

Std. Predicted Value -2.378 2.345 .000 1.000 40

Std. Residual -1.988 2.320 -.028 1.000 40

a. Dependent Variable: FDY

b. Linear Regression through the Origin

Coefficientsa,b

Model

Unstandardized Coefficients

Standardized

Coefficients

t Sig.B Std. Error Beta

1 FDX .720 .077 .831 9.328 .000

a. Dependent Variable: FDY

b. Linear Regression through the Origin

The results have improved, especially in terms of DW statistic, which is now = 1.611.

Since no-autocorrelation zone ranges between du and 4-du, that is:

du = 1.544 and 4 – du = 4 – 1.544 = 2.456

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The calculated DW = 1.611 falls within the no-autocorrelation zone, suggesting that there exists

no autocorrelation problem, now.

The Rho-Corrected regression

Where the ‘coefficient of autocorrelation’ (rho = ρ) is known, or can be estimated, the value of

the ρ is used for correction of autocorrelation in the following form.

(Yt – ρYt-1) = α + β1(Xt – ρXt-1) + et (9.9)

The coefficient of autocorrelation ρ (rho) can be calculated, using the estimated DW statistic, as

follows.

DW = d = 2(1 – ρ) (9.10a)

ρ = 1 – (d/2) (9.10b)

In our original model (9.2), DW estimates at 0.123; putting this value in 9.10b:

ρ = 1 – (d/2) (9.11a)

= 1 – (0.123/2)

= 1 – 0.0615

= 0.9385 (9.11b)

Substituting ρ = 0.9385 in (9.9),

(Yt – 0.9385Yt-1) = α + β1(Xt – 0.9385Xt-1) + et (9.12)

and running the regression.

Prais-Winsten transformation: In case of the use of both cases of the First-differencing or the

Rho-Corrected regression, the first observation, because of not having any antecedent is lost; in

such situation, Prais-Winsten transformation helps to make good of this loss. According to this

transformation, the first observation can be retained after transforming it in the following way.

Y1 √(1 – ρ2) and Y1√(1 – ρ2) (9.13)

The correction of Autocorrelation through the use of First-differencing or Rho-corrected

regression is referred generally referred to as Generalized Least Square (GLS); when instead o

true ρ, estimated ρ is used, the method is known as Feasible GLS (FGLS) or Estimated GLS

(EGLS). In case, GLS is used with Prais-Winsten transformation, method is then called Full

EGLS or FEGLS (Gujarati 2007, pp.487-494).

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The Heteroscadasticity-and-autocorrelation consistent standard errors (HAC)

Instead of using the FGLS methods discussed earlier, one can use OLS after correcting standard

errors for autocorrelation the procedure developed by Newey and West15 This method is an

extension of White’s heteroscedasticity-consistent standard errors discussed earlier under

Heteroscadasticilty. The corrected standard errors are known as HAC (heteroscedasticity- and

autocorrelation-consistent) standard errors or simply as Newey–West standard errors. Most

modern computer packages now calculate the Newey–West standard errors. However, it is

important to point out that the Newey–West procedure is strictly speaking valid in large samples

and may not be appropriate in small samples. Therefore, if a sample is reasonably large, one

should use the Newey–West procedure to correct OLS standard errors not only in situations of

autocorrelation only but also in cases of heteroscedasticity, for the HAC method can handle both,

unlike the White method, which was designed specifically for heteroscedasticity (Gujarati 2007,

pp.494-95)

OLS versus FGLS and HAC

In the presence of autocorrelation, OLS estimators, although unbiased, consistent, and

asymptotically normally distributed, are not efficient. Therefore, the usual inference procedure

based on the t, F, and χ2 tests is no longer appropriate. On the other hand, FGLS and HAC

produce estimators that are efficient, but the finite, or small-sample, properties of these

estimators are not well documented. This means in small samples the FGLS and HAC might

actually do worse than OLS. As a matter of fact, in a Monte Carlo study Griliches and

Rao46 found that if the sample is relatively small and the coefficient of autocorrelation,

ρ, is less than 0.3, OLS is as good or better than FGLS. As a practical matter, then, one may use

OLS in small samples in which the estimated Rho is, say, less than 0.3 (Gujarati 2007, p,495).

15 W. K. Newey, and K. West, “A Simple Positive Semi-Definite Heteroscedasticity and AutocorrelationConsistent Covariance Matrix, Econometrica, vol. 55, 1987, pp. 703–708.

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TOPICS 10 – 15SPECIAL APPLICATIONS

Topic 10Mediation analysis: problems and prospects

Topic 11Moderation analysis: problems and prospects

Topics 12 - 13Time-series analysis: problems and prospects

Topic 14Panel data analysis: problems and prospects

Topic 15Minimization, maximization and optimization

Topic 16Welfare analysis: maximization of producer and consumer

surpluses and minimization of social costs

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Aqt instructor-notes-final

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