Quantity Demand Analysis

MMT BATCH 36*QUANTITY DEMAND ANALYSISJoseph Winthrop B. Godoy

MMT BATCH 36

INTRODUCTIONShows how a manager can use elasticities of demand as a quantitative forecasting tool

Describes regression analysis, which is the technique economists use to estimate the parameters of demand functions

THE ELASTICITY CONCEPTElasticity AnalysisThe primary tool used to determine the magnitude of such a change

Elasticity Measures the responsiveness of one variable to changes in another variable

THE ELASTICITY CONCEPTTwo aspects of Elasticity

(1) Whether it is positive or negative(2) Whether it is greater than 1 or less than 1 in absolute value

OWN PRICE ELASTICITY OF DEMAND

measures the responsiveness of quantity demanded to a change in price;

the percentage change in quantity demanded divided by the percentage change in the price of the good

ELASTIC DEMAND

demand is said to be elastic if the absolute value of the own price elasticity is greater than 1:

INELASTIC DEMAND

demand is said to be inelastic if the absolute value of the own price elasticity is less than 1:

UNITARY ELASTIC

if the absolute value of the own price elasticity is equal to 1:

ELASTICITY & TOTAL REVENUE

Price of Software Quantity of Software SoldOwn Price ElasticityTotal RevenueA$ 0800.00$ 0B 570-0.14350C 1060-0.33600D 1550-0.60750E 2040-1.00800F 2530-1.67750G 3020-3.00600H 3510-7.00350I 4000

TOTAL REVENUE TEST

If demand is elastic, an increase (decrease) in price will lead to a decrease (increase) in total revenue. If demand is inelastic, an increase (decrease) in price will lead to an increase (decrease) in total revenue. Finally, total revenue is maximized at the point where demand is unitary elastic.

THE ELASTICITY CONCEPTPerfectly ElasticIf the own price elasticity of demand is indefinite on absolute value

Perfectly InelasticIf the own price elasticity of demand is zero

FACTORS AFFECTING THE OWN PRICE ELASTICITY

FACTORS AFFECTING THE OWN PRICE ELASTICITYAvailable Substitutes2. Time3. Expenditure Share

FACTORS AFFECTING THE OWN PRICE ELASTICITYAvailable SubstitutesOne key determinant of the elasticity of demand for a good is the number of close substitutes for the good.

The more substitutes available for the good, the more elastic the demand for it

FACTORS AFFECTING THE OWN PRICE ELASTICITYTime The more time consumers have to react to a price change, the more elastic the demand for the good

Time allows the consumer to seek out available substitutes

FACTORS AFFECTING THE OWN PRICE ELASTICITYExpenditure ShareGoods that comprise a relatively small share of consumers budgets tend to be more inelastic than goods for which consumers spend a sizable portion of their incomes.

When a good comprises only a small portion of the budget, the consumer can reduce the consumption of other goods when the price of the good increases.

MARGINAL REVENUE AND THE OWN PRICE ELASTICITY OF DEMAND Marginal RevenueThe change in total revenue due to a change in output, and that to maximize profits, a firm should produce where marginal revenue equals marginal cost.

Cross-Price Elasticity

Cross-Price ElasticityA measure of the responsiveness of the demand for a good to changes in the price of a related good: the percentage change in the quantity demanded of one good divided by the percentage change in the price of a related good.

Whenever goods X and Y are substitutes, an increase in the price of Y leads to an increase in the demand for X.When goods X and Y are complements, an increase in the price of Y leads to a decrease in the demand for X.

ExampleIf the cross-price elasticity of demand between Corel WordPerfect and Microsoft Word processing software is 3, a 10% hike in the price of Word will increase the demand for WordPerfect by 30 percent, since 30%/10% = 3.

This demand increase for WordPerfect occurs because consumers substitute away from Word and toward WordPerfect, due to the price increase.

Cross-Price ElasticityCross-price elasticities play an important role in the pricing decisions of firms that sell multiple products.

ExampleIn fastfood chains, hamburgers and sodas are complements. When customers buy hamburgers, they buy sodas as well. If the fastfood chain decides to lower the price on hamburgers, the fastfood chains revenues from both hamburgers and sodas are affected. In addition, reducing the price of hamburgers increases the quantity demanded on sodas, thus increasing soda revenues.

Income ElasticityIncome Elasticity is a measure of the responsiveness of consumer demand to changes in income.

Income ElasticityWhen good X is a normal good, an increase in income leads to an increase in the consumption of X. When X is an inferior good, an increase in income leads to a decrease in the consumption of X.

Income ElasticityThe formula forincomeelasticityis:IncomeElasticity= (% change in quantity demanded) / (% change inincome)

Example 1An example of a product with positive incomeelasticitycould be Ferraris. Let's say theeconomyis booming and everyone's income rises by 400%. Because people have extramoney, the quantity of Ferraris demanded increases by 15%.We can use the formula to figure out the income elasticity forthis Italian sports car:Income Elasticity = 15% / 400% = 0.0375

Example 2An example of a good with negative income elasticity could be cheap shoes. Let's again assume the economy is doing well and everyone's income rises by 30%. Because people have extra money and can afford nicer shoes, the quantity of cheap shoes demanded decreases by 10%.The income elasticity of cheap shoes is:Income Elasticity = -10% / 30% = -0.33

Log-Linear DemandDemand is log-linear if the logarithm of demand is a linear function of the logarithms of prices, income, and other variables.

MMT BATCH 36*ECONOMETRICS & REGRESSION ANALYSISJoseph Winthrop B. Godoy

MMT BATCH 36

MMT BATCH 36*EconometricsJoseph Winthrop B. Godoy

MMT BATCH 36

MMT BATCH 36*IntroductionManagers may obtain estimates of demand and elasticity from published studies available in the library or from a consultant hired to estimate the demand function based on the specifics of their company product.The primary job of a manager is to use the information to make decisions

MMT BATCH 36

MMT BATCH 36*IntroductionRegardless of how the manager obtains the estimates, it is useful to have a general understanding of how demand functions are estimated and what the various diagnostic statistics that accompany the reported output mean. This entails knowledge of a branch of economics called econometrics. Econometrics is simply the statistical analysis of economic phenomena.

MMT BATCH 36

EconometricsLets briefly examine the basic ideas underlying the estimation of the demand for a product. Suppose there is some underlying data on the relation between a dependent variable, Y, and some explanatory variable, X. Suppose that when the values of X and Y are plotted, they appear as points A, B, C, D, E, and F in Figure 34.

MMT BATCH 36*

MMT BATCH 36

EconometricsMMT BATCH 36*

MMT BATCH 36

EconometricsClearly, the points do not lie on a straight line, or even a smooth curve (try alternative ways of connecting the dots if you are not convinced). The job of the econometrician is to find a smooth curve or line that does a good job of approximating the points.

MMT BATCH 36*

MMT BATCH 36

EconometricsFor example, suppose the econometrician believes that, on average, there is a linear relation between Y and X, but there is also some random variation in the relationship. Mathematically, this would imply that the true relationship between Y and X is

Y = a + bX + eMMT BATCH 36*

MMT BATCH 36

Econometricswhere a and b are unknown parameters and e is a random variable (an error term) that has a zero mean. Because the parameters that determine the expected relation between Y and X are unknown, the econometrician must find out the values of the parameters a and b.

MMT BATCH 36*

MMT BATCH 36

EconometricsNote that for any line drawn through the points, there will be some discrepancy between the actual points and the line. For example, consider the line in slide 6 or the Figure 34, which does a reasonable job of fitting the data.

MMT BATCH 36*

MMT BATCH 36

EconometricsIf a manager used the line to approximate the true relation, there would be some discrepancy between the actual data and the line. For example, points A and D actually lie above the line, while points C and E lie below it.

MMT BATCH 36*

MMT BATCH 36

EconometricsThe deviations between the actual points and the line are given by the distance of the dashed lines in Figure 34, namely A, C, D, and E. Since the line represents the expected, or average, relation between Y and X, these deviations are analogous to the deviations from the mean used to calculate the variance of a random variable.

MMT BATCH 36*

MMT BATCH 36

EconometricsThe econometrician uses a regression software package to find the values of a and b that minimize the sum of the squared deviations between the actual points and the line. In essence, the regression line is the line that minimizes the squared deviations between the line (the expected relation) and the actual data points. These values of a and b, which frequently are denoted and b, are called parameter estimates, and the corresponding line is called the least squares regression.

MMT BATCH 36*

MMT BATCH 36

Regression Output in Excel

Data

Heating OilTemperatureInsulation

275.3403

363.8273

164.34010Data SheetData for developing a regression model in order to predict the

40.8736consumption of heating oil by single-family homes during January.

94.3646VariableRangeValues

230.9346Heating oilA2:A16(in gallons)

366.796TemperatureB2:B16(average daily, in degrees F)

300.6810InsulationC2:C16(in inches)

237.82310

121.4633

31.46510

203.5416

441.1213

323.0383

52.55810

Correlation

Heating OilTemperatureInsulation

Heating Oil1TemperatureHeating OilInsulationHeating Oil

Temperature-0.8697411704140275.33275.3

Insulation-0.46508252720.0089220394127363.83363.8

40164.310164.3

7340.8640.8

6494.3694.3

34230.96230.9

9366.76366.7

8300.610300.6

23237.810237.8

63121.43121.4

6531.41031.4

41203.56203.5

21441.13441.1

38323.03323.0

5852.51052.5

Correlation

Heating Oil Vs. Temperature

Temperature

Heating Oil

regr

Heating Oil

Insulation

Heating Oil

SUMMARY OUTPUT

Regression Statistics

Multiple R0.9826547566

R Square0.9656103706

Adjusted R Square0.9598787657

Standard Error26.0137832312

Observations15

ANOVA

dfSSMSFSignificance F

Regression2228014.62631736114007.31315868168.47120284210.0000000017

Residual128120.6030159735676.7169179978

Total14236135.229333333

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%

Intercept562.151009228521.093104328626.65093769370516.1930836858608.1089347712

Temperature-5.4365805880.3362161666-16.16989641960.0000000016-6.1691326727-4.7040285033

Insulation-20.01232066622.3425052266-8.54312743430.0000019073-25.1162010201-14.9084403122

Correlation coefficients measure the magnitude and direction of the linear relationship between any two variables. You want high correlation between independent and dependent variables, and low correlations among independent variables. Check the two scatter plots below that demonstrate the correlation idea.

magnitude and direction of the linear relationship between two variables.

Estimated Heating Oil = 562.15 - 5.436 (Temperature) - 20.012 (Insulation)

Y = B0 + B1 X1 + B2X2 + B3X3 - - - +/- ErrorTotal = Estimated/Predicted +/- Error

Correlation between actual y and predicted y

Proportion of variance in Y that is explained by the model (independent variables)

Adjusted for number of independent variables. Useful when comparing different models (with different or different number of independent variables.

Measure of variability around the regression line - (difference between observed and predicted values)

degrees of freedom

Sum of Squared deviations from the mean

Number of Independent Variables (k). DFR

Sum of squared deviations of predicted values from the mean.Regression Sum of Squares (SSR)

Mean SS= SSR / dfRMeasure of variance of the predicted part

Variance ratio.Indicates the variance explained by the model per unit error

p-Value of FProbability of Type I error (rejecting a true null)Null: None of the independent variables is a significant predictor of y. (All coefficients = 0)

Error degrees of freedom= (n-1) - kDFE

Difference between total and regression sum of squares of error (SSE)

Mean SS of Error= SSE / DFEMeasure of Error Variance

Total degrees of freedom (n-1)DFT

Sum of squared deviations of actual y form the mean. Total Sum of Squares (SST)

Estimate of the coefficient of the variable in the regression equation

Standard error of the coefficient

Coefficient devided by its standard error

Probability of Type I error (of rejecting a true null).Null: The variable has no impact on the dependent variable (b=0)

lower limit of 95% confidence interval of estimate

upperlimit of 95% confidence interval of estimate

MMT BATCH 36*

MMT BATCH 36

MMT BATCH 36*Regression AnalysisJoseph Winthrop B. Godoy

MMT BATCH 36

MMT BATCH 36*IntroductionMany problems in engineering and science involve exploring the relationships between two or more variables. Regression analysis is a statistical technique that is very useful for these types of problems. For example, in a chemical process, suppose that the yield of the product is related to the process-operating temperature. Regression analysis can be used to build a model to predict yield at a given temperature level.

MMT BATCH 36

MMT BATCH 36*Regression AnalysisRegression Analysis: the study of the relationship between variables

Regression Analysis: one of the most commonly used tools for business analysis

Easy to use and applies to many situations

MMT BATCH 36

MMT BATCH 36*Regression Modeling PhilosophyNature of the relationshipsModel Building Procedure

Determine dependent variable (y)Determine potential independent variable (x)Collect relevant dataHypothesize the model formFitting the modelDiagnostic check: test for significance

MMT BATCH 36

Basic idea:Use data to identify relationships among variables and use these relationships to make predictions

MMT BATCH 36*

MMT BATCH 36

Linear RegressionFocus:

Gain some understanding of the mechanics. the regression line regression error Learn how to interpret and use the results. Learn how to setup a regression analysis.MMT BATCH 36*

MMT BATCH 36

Linear RegressionRegression is the attempt to explain the variation in a dependent variable using the variation in independent variables.Regression is thus an explanation of causation.If the independent variable(s) sufficiently explain the variation in the dependent variable, the model can be used for prediction.

Independent variable (x)Dependent variable

Linear RegressionLinear dependence: constant rate of increase of one variable with respect to another (as opposed to, e.g., diminishing returns).Regression analysis describes the relationship between two (or more) variables.Examples:

Income and educational level Demand for electricity and the weather Home sales and interest ratesMMT BATCH 36*

MMT BATCH 36

MMT BATCH 36*Regression AnalysisSimple Regression: single explanatory variable

Multiple Regression: includes any number of explanatory variables.

MMT BATCH 36

Single RegressionMultiple Regression

MMT BATCH 36*Regression AnalysisLinear Regression: straight-line relationship

Form: y = mx + b (linear equation)

Non-linear: implies curved relationships, for example logarithmic or curvilinear relationships

MMT BATCH 36

Scatter plotsRegression analysis requires interval and ratio-level data.To see if your data fits the models of regression, it is wise to conduct a scatter plot analysis.The reason?

Regression analysis assumes a linear relationship. If you have a curvilinear relationship or no relationship, regression analysis is of little use.

Scatter plotThis is a linear relationshipIt is a positive relationship.As population with BAs increases so does the personal income per capita.

Regression LineRegression line is the best straight line description of the plotted points and can use it to describe the association between the variables.If all the lines fall exactly on the line then the line is 0 and you have a perfect relationship.

Types of Lines

Scatter Plots of Data with Various Correlation Coefficients

Y

X

YX

Y

X

Y

X

Y

Xr = -1r = -.6r = 0r = +.3r = +1

YXr = 0

YX

Y

X

Y

Y

XXLinear relationshipsCurvilinear relationshipsLinear Correlation

YX

YX

Y

Y

XXStrong relationshipsWeak relationships

Linear Correlation

Linear Correlation

YX

YX

No relationship

Things to remember Regressions are still focuses on association, not causation.Association is a necessary prerequisite for inferring causation, but also:

The independent variable must preceded the dependent variable in time.The two variables must be plausibly lined by a theory,Competing independent variables must be eliminated.

Regression TableThe regression coefficient is not a good indicator for the strength of the relationship.Two scatter plots with very different dispersions could produce the same regression line.

Simple Linear Regression

Independent variable (x)Dependent variable (y)

The output of a regression is a function that predicts the dependent variable based upon values of the independent variables.Simple regression fits a straight line to the data.y = b0 + b1X b0 (y intercept)

b1 = slope= y/ x



The function will make a prediction for each observed data point. The observation is denoted by y and the prediction is denoted by y.

ZeroPrediction: yObservation: y^^

Simple Linear RegressionFor each observation, the variation can be described as: y = y + Actual = Explained + Error

ZeroPrediction error: ^Prediction: y ^

Observation: y

SIMPLE REGRESSIONMMT BATCH 36*

Relationship

YXDependent VariableIndependent Variable

y = mx + bLinear Equation

= b0 + b1xb0 = y-interceptb1 = slopeb0 = - b1

MMT BATCH 36


VARIABLE DATAXYX2Y2XY1214224416835925154716492858256440SXSYSX2SY2SXY15265515893 35.2n =5

MMT BATCH 36

SIMPLE REGRESSION

MMT BATCH 36*

MMT BATCH 36


MMT BATCH 36

Calculating SSE


The line that minimizes the sum of squared deviations between the line and the actual data points is the least squares regression.A least squares regression selects the line with the lowest total sum of squared prediction errors. This value is called the Sum of Squares of Error, or SSE(2)

Calculating SSR


The Sum of Squares Regression (SSR) is the sum of the squared differences between the prediction for each observation and the population mean.Population mean: y

Regression FormulasCalculating SSTThe Total Sum of Squares (SST) is equal to SSR + SSE.Mathematically,

SSR = ( y y ) (measure of explained variation)

SSE = ( y y ) (measure of unexplained variation)

SST = SSR + SSE = ( y y ) (measure of total variation in y)MMT BATCH 36*

MMT BATCH 36

Regression CoefficientThe regression coefficient is the slope of the regression line and tells you what the nature of the relationship between the variables is.How much change in the independent variables is associated with how much change in the dependent variable.The larger the regression coefficient the more change.

The Coefficient of Determination

Standard Error of RegressionThe Standard Error of a regression is a measure of its variability. It can be used in a similar manner to standard deviation, allowing for prediction intervals.y 2 standard errors will provide approximately 95% accuracy, and 3 standard errors will provide a 99% confidence interval.Standard Error is calculated by taking the square root of the average prediction error.Standard Error = SSEn-kWhere n is the number of observations in the sample and k is the total number of variables in the model

The output of a simple regression is the coefficient and the constant A. The equation is then:y = A + * x + where is the residual error. is the per unit change in the dependent variable for each unit change in the independent variable. Mathematically:

= y x

Multiple Linear RegressionMore than one independent variable can be used to explain variance in the dependent variable, as long as they are not linearly related.

A multiple regression takes the form:

y = A + X + X + + k Xk +

where k is the number of variables, or parameters. 1 1 2 2

MulticollinearityMulticollinearity is a condition in which at least 2 independent variables are highly linearly correlated. It will often crash computers.A correlations table can suggest which independent variables may be significant. Generally, an ind. variable that has more than a .3 correlation with the dependent variable and less than .7 with any other ind. variable can be included as a possible predictor.

Example table of CorrelationsYX1X2Y1.000X10.8021.000X20.8480.5781.000

Nonlinear Regression

Nonlinear functions can also be fit as regressions. Common choices include Power, Logarithmic, Exponential, and Logistic, but any continuous function can be used.

Some Aplications

MMT BATCH 36*

MMT BATCH 36

Sample 15 houses from the region.

House NumberY: Actual Selling Price ($1,000s)X: House Size (100s ft2)189.520.0279.914.8383.120.5456.912.5566.618.0682.514.37126.327.5879.316.59119.924.31087.620.211112.622.012120.8.0191378.512.31474.314.01574.816.7Averages88.8418.17

MMT BATCH 36*Simple Regression Modely = a + bx + e (Note: y = mx + b)Coefficients: a and bVariable a is the y intercept Variable b is the slope of the line

MMT BATCH 36

MMT BATCH 36*Simple Regression ModelPrecision: accepted measure of accuracy is mean squared errorAverage squared difference of actual and forecast

MMT BATCH 36

MMT BATCH 36*Simple Regression ModelAverage squared difference of actual and forecastSquaring makes difference positive, and severity of large errors is emphasized

MMT BATCH 36

MMT BATCH 36*Simple Regression ModelError (residual) is difference of actual data point and the forecasted value of dependant variable y given the explanatory variable x.

Error

MMT BATCH 36

MMT BATCH 36*Simple Regression Model y = mx + b

Y= a + bX + e = 56,104 + 63.11(Sq ft) + e

If X = 2,500 Square feet, then

$213,879 = 56,104 + 63.11(2,500)

MMT BATCH 36

MMT BATCH 36*Simple Regression ModelLinearity

MMT BATCH 36

Chart2

231000238178.237070537

170000196840.681588645

217000206117.965032673

209000217099.239313359

218000219686.78095421

218000266956.748978541

310000222211.211823333

248000247581.742058021

186000185606.964221047

204000186490.515025241

185000194505.583034707

216000207064.626608594

146000186111.850394872

293000274277.598498998

234000210725.051368823

219000202709.983359357

267000263043.8811314

297000264874.093511515

192000191160.712133118

217000184155.416471302

234000237799.572440169

177000208074.398956244

234000259825.231773268

249000217667.236258912

206000213880.589955227

112000197976.675479751

283000226250.30121393

153000207569.512782419

251000194694.915349891

191000237862.683211897

Cost

Predicted Cost

Square Feet

Cost

Square Feet Line Fit Plot

Source

Square FeetCost

2,885231,000

2,230170,000

2,377217,000

2,551209,000

2,592218,000

3,341218,000

2,632310,000

3,034248,000

2,052186,000

2,066204,000

2,193185,000

2,392216,000

2,060146,000

3,457293,000

2,450234,000

2,323219,000

3,279267,000

3,308297,000

2,140192,000

2,029217,000

2,879234,000

2,408177,000

3,228234,000

2,560249,000

2,500206,000

2,248112,000

2,696283,000

2,400153,000

2,196251,000

2,880191,000

Residuals

SUMMARY OUTPUT



R Square0.35885033



Observations30

ANOVA


Regression121313807694.469921313807694.469915.67155020050.000469324

Residual2838080892305.53011360031868.05465

Total2959394700000

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

Intercept56103.660635029341670.92073372391.34635039610.1889884842-29255.446855601141462.76812566-29255.446855601141462.76812566

Square Feet63.110771728115.94217278223.95873088260.00046932430.454674473895.766868982330.454674473895.7668689823

RESIDUAL OUTPUT

ObservationPredicted CostResidualsStandard Residuals

1238178.237070537-7178.2370705372-0.198090313

2196840.681588645-26840.6815886455-0.7406942631

3206117.96503267310882.03496732690.3003001561

4217099.239313359-8099.2393133588-0.2235062502

5219686.78095421-1686.7809542101-0.0465483327

6266956.748978541-48956.7489785412-1.3510082815

7222211.21182333387788.78817666682.4226155194

8247581.742058021418.2579419790.0115422277

9185606.964221047393.03577895260.010846198

10186490.51502524117509.48497475950.4831909737

11194505.583034707-9505.5830347065-0.2623156494

12207064.6266085948935.37339140570.2465801693

13186111.850394872-40111.850394872-1.1069248511

14274277.59849899818722.40150100160.5166625645

15210725.05136882323274.94863117720.642294454

16202709.98335935716290.01664064320.4495385794

17263043.88113143956.11886859970.1091728815

18264874.09351151532125.90648848540.8865450959

19191160.712133118839.28786688160.0231609509

20184155.41647130232844.58352869840.9063776757

21237799.572440169-3799.5724401687-0.1048528332

22208074.398956244-31074.3989562435-0.8575277405

23259825.231773268-25825.2317732683-0.7126719549

24217667.23625891231332.76374108850.8646575637

25213880.589955227-7880.5899552268-0.2174724122

26197976.675479751-85976.6754797509-2.3726085375

27226250.3012139356749.69878606971.5660621801

28207569.512782419-54569.5127824189-1.5058978635

29194694.91534989156305.08465010921.553792628

30237862.683211897-46862.6832118968-1.2932205352

Residuals

2885

2230

2377

2551

2592

3341

2632

3034

2052

2066

2193

2392

2060

3457

2450

2323

3279

3308

2140

2029

2879

2408

3228

2560

2500

2248

2696

2400

2196

2880

Square Feet

Residuals

Square Feet Residual Plot

Final

2310002885

1700002230

2170002377

2090002551

2180002592

2180003341

3100002632

2480003034

1860002052

2040002066

1850002193

2160002392

1460002060

2930003457

2340002450

2190002323

2670003279

2970003308

1920002140

2170002029

2340002879

1770002408

2340003228

2490002560

2060002500

1120002248

2830002696

1530002400

2510002196

1910002880

Cost

Predicted Cost

Square Feet

Cost


Sheet3

Square FeetCost

2,885231,000Square FeetCost

2,230170,000Square Feet1

2,377217,000Cost0.59901

2,551209,000

2,592218,0000.5990

3,341218,000

2,632310,000

3,034248,000

2,052186,000

2,066204,000

2,193185,000

2,392216,000

2,060146,000

3,457293,000

2,450234,000

2,323219,000

3,279267,000

3,308297,000

2,140192,000

2,029217,000

2,879234,000

2,408177,000SUMMARY OUTPUT

3,228234,000

2,560249,000Regression Statistics

2,500206,000Multiple R0.599

2,248112,000R Square0.359

2,696283,000Adjusted R Square0.336

2,400153,000Standard Error36,878.61

2,196251,000Observations30

2,880191,000

Square FeetCostANOVA


Mean2,579.53Mean218,900Regression121313807694.469921313807694.469915.67155020050.000469324

Standard Error78.43Standard Error8,263Residual2838080892305.53011360031868.05465

Median2,475.00Median217,500Total2959394700000

ModeMode234,000

Standard Deviation429.56Standard Deviation45,256CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

Sample Variance184,525.36Sample Variance2,048,093,103Intercept56,103.6641,670.921.350.1890(29,255.45)141,462.77(29,255.45)141,462.77

Kurtosis-0.69Kurtosis0.22Square Feet63.1115.943.960.000530.4595.7730.4595.77

Skewness0.64Skewness-0.04

Range1,428.00Range198,000

Minimum2,029.00Minimum112,000

Maximum3,457.00Maximum310,000

Sum77,386.00Sum6,567,000

Count30.00Count30

Bin

1500

2000

2500

3000

3500

Sheet3

Scatter Plot

Square Footage

Cost

Scatter Plot of Housing Cost/Sqaure Foot

MMT BATCH 36*Simple Regression ModelLinearity

MMT BATCH 36

Chart1

-7178.2370705372

-26840.6815886455

10882.0349673269

-8099.2393133588

-1686.7809542101

-48956.7489785412

87788.7881766668

418.257941979

393.0357789526

17509.4849747595

-9505.5830347065

8935.3733914057

-40111.850394872

18722.4015010016

23274.9486311772

16290.0166406432

3956.1188685997

32125.9064884854

839.2878668816

32844.5835286984

-3799.5724401687

-31074.3989562435

-25825.2317732683

31332.7637410885

-7880.5899552268

-85976.6754797509

56749.6987860697

-54569.5127824189

56305.0846501092

-46862.6832118968

Square Feet

Residuals


Source

Square FeetCost

2,885231,000

2,230170,000

2,377217,000

2,551209,000

2,592218,000

3,341218,000

2,632310,000

3,034248,000

2,052186,000

2,066204,000

2,193185,000

2,392216,000

2,060146,000

3,457293,000

2,450234,000

2,323219,000

3,279267,000

3,308297,000

2,140192,000

2,029217,000

2,879234,000

2,408177,000

3,228234,000

2,560249,000

2,500206,000

2,248112,000

2,696283,000

2,400153,000

2,196251,000

2,880191,000

Residuals

SUMMARY OUTPUT



R Square0.35885033



Observations30

ANOVA


Regression121313807694.469921313807694.469915.67155020050.000469324

Residual2838080892305.53011360031868.05465

Total2959394700000

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

Intercept56103.660635029341670.92073372391.34635039610.1889884842-29255.446855601141462.76812566-29255.446855601141462.76812566

Square Feet63.110771728115.94217278223.95873088260.00046932430.454674473895.766868982330.454674473895.7668689823

RESIDUAL OUTPUT

ObservationPredicted CostResidualsStandard Residuals

1238178.237070537-7178.2370705372-0.198090313

2196840.681588645-26840.6815886455-0.7406942631

3206117.96503267310882.03496732690.3003001561

4217099.239313359-8099.2393133588-0.2235062502

5219686.78095421-1686.7809542101-0.0465483327

6266956.748978541-48956.7489785412-1.3510082815

7222211.21182333387788.78817666682.4226155194

8247581.742058021418.2579419790.0115422277

9185606.964221047393.03577895260.010846198

10186490.51502524117509.48497475950.4831909737

11194505.583034707-9505.5830347065-0.2623156494

12207064.6266085948935.37339140570.2465801693

13186111.850394872-40111.850394872-1.1069248511

14274277.59849899818722.40150100160.5166625645

15210725.05136882323274.94863117720.642294454

16202709.98335935716290.01664064320.4495385794

17263043.88113143956.11886859970.1091728815

18264874.09351151532125.90648848540.8865450959

19191160.712133118839.28786688160.0231609509

20184155.41647130232844.58352869840.9063776757

21237799.572440169-3799.5724401687-0.1048528332

22208074.398956244-31074.3989562435-0.8575277405

23259825.231773268-25825.2317732683-0.7126719549

24217667.23625891231332.76374108850.8646575637

25213880.589955227-7880.5899552268-0.2174724122

26197976.675479751-85976.6754797509-2.3726085375

27226250.3012139356749.69878606971.5660621801

28207569.512782419-54569.5127824189-1.5058978635

29194694.91534989156305.08465010921.553792628

30237862.683211897-46862.6832118968-1.2932205352

Residuals

2885

2230

2377

2551

2592

3341

2632

3034

2052

2066

2193

2392

2060

3457

2450

2323

3279

3308

2140

2029

2879

2408

3228

2560

2500

2248

2696

2400

2196

2880

Square Feet

Residuals


Final

2310002885

1700002230

2170002377

2090002551

2180002592

2180003341

3100002632

2480003034

1860002052

2040002066

1850002193

2160002392

1460002060

2930003457

2340002450

2190002323

2670003279

2970003308

1920002140

2170002029

2340002879

1770002408

2340003228

2490002560

2060002500

1120002248

2830002696

1530002400

2510002196

1910002880

Cost

Predicted Cost

Square Feet

Cost


Sheet3

Square FeetCost

2,885231,000Square FeetCost

2,230170,000Square Feet1

2,377217,000Cost0.59901

2,551209,000

2,592218,0000.5990

3,341218,000

2,632310,000

3,034248,000

2,052186,000

2,066204,000

2,193185,000

2,392216,000

2,060146,000

3,457293,000

2,450234,000

2,323219,000

3,279267,000

3,308297,000

2,140192,000

2,029217,000

2,879234,000

2,408177,000SUMMARY OUTPUT

3,228234,000

2,560249,000Regression Statistics

2,500206,000Multiple R0.599

2,248112,000R Square0.359

2,696283,000Adjusted R Square0.336

2,400153,000Standard Error36,878.61

2,196251,000Observations30

2,880191,000

Square FeetCostANOVA


Mean2,579.53Mean218,900Regression121313807694.469921313807694.469915.67155020050.000469324

Standard Error78.43Standard Error8,263Residual2838080892305.53011360031868.05465

Median2,475.00Median217,500Total2959394700000

ModeMode234,000

Standard Deviation429.56Standard Deviation45,256CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

Sample Variance184,525.36Sample Variance2,048,093,103Intercept56,103.6641,670.921.350.1890(29,255.45)141,462.77(29,255.45)141,462.77

Kurtosis-0.69Kurtosis0.22Square Feet63.1115.943.960.000530.4595.7730.4595.77

Skewness0.64Skewness-0.04

Range1,428.00Range198,000

Minimum2,029.00Minimum112,000

Maximum3,457.00Maximum310,000

Sum77,386.00Sum6,567,000

Count30.00Count30

Bin

1500

2000

2500

3000

3500

Sheet3

Scatter Plot

Square Footage

Cost

Scatter Plot of Housing Cost/Sqaure Foot

MMT BATCH 36*Simple Regression ModelIndependence:

Errors must not correlateTrials must be independent

MMT BATCH 36

MMT BATCH 36*Simple Regression ModelHomoscedasticity:

Constant varianceScatter of errors does not change from trial to trialLeads to misspecification of the uncertainty in the model, specifically with a forecastPossible to underestimate the uncertaintyTry square root, logarithm, or reciprocal of y

MMT BATCH 36

MMT BATCH 36*Simple Regression ModelNormality:

Errors should be normally distributedPlot histogram of residuals

MMT BATCH 36

MMT BATCH 36*Multiple Regression ModelY = + 1X1 + + kXk +

MMT BATCH 36

Example: An Empirical Model MMT BATCH 36*

MMT BATCH 36

Empirical Model

Figure 1 Scatter Diagram of oxygen purity versus hydrocarbon level from Table 11-1.

Empirical Model

Based on the scatter diagram, it is probably reasonable to assume that the mean of the random variable Y is related to x by the following straight-line relationship:where the slope and intercept of the line are called regression coefficients.The simple linear regression model is given bywhere is the random error term.

Empirical Models

We think of the regression model as an empirical model.Suppose that the mean and variance of are 0 and 2, respectively, thenThe variance of Y given x is

Empirical Models

The true regression model is a line of mean values:

where 1 can be interpreted as the change in the mean of Y for a unit change in x. Also, the variability of Y at a particular value of x is determined by the error variance, 2. This implies there is a distribution of Y-values at each x and that the variance of this distribution is the same at each x.

Empirical Models

Figure 2 The distribution of Y for a given value of x for the oxygen purity-hydrocarbon data.


The case of simple linear regression considers a single regressor or predictor x and a dependent or response variable Y. The expected value of Y at each level of x is a random variable:

We assume that each observation, Y, can be described by the model


Suppose that we have n pairs of observations (x1, y1), (x2, y2), , (xn, yn).

Figure 3 Deviations of the data from the estimated regression model.


The method of least squares is used to estimate the parameters, 0 and 1 by minimizing the sum of the squares of the vertical deviations in Figure 3.

Figure 3 Deviations of the data from the estimated regression model.


Using the following Equation, the n observations in the sample can be expressed as

The sum of the squares of the deviations of the observations from the true regression line is

Quantity Demand Analysis

Economy & Finance