Econ 240 C Lecture 15
Jan 25, 2016
Econ 240 C
Lecture 15
2
Outline Project II Forecasting ARCH-M Models Granger Causality Simultaneity VAR models
3I. Work in GroupsII. You will be graded based on a PowerPoint presentation and a written report.III. Your report should have an executive summary of one to one and a half pages that summarizes your findings in words for a non-technical reader. It should explain the problem being examined from an economic perspective, i.e. it should motivate interest in the issue on the part of the reader. Your report should explain how you are investigating the issue, in simple language. It should explain why you are approaching the problem in this particular fashion. Your executive report should explain the economic importance of your findings.
4
Technical Appendix1. Table of Contents2. Spreadsheet of data used and sources or, if extensive, a subsample of the data3. Describe the analytical time series techniques you are using4. Show descriptive statistics and histograms for the variables in the study5. Use time series data for your project; show a plot of each variable against time
The technical details of your findings you can attach as an appendix
5Group A Group BGroup C
Eirk Skeid Markus Ansmann Xiaoyin ZhangTor Seim Theresa FirestineSamantha GardnerBradley Moore Nikolay Laptev Ryan NabingerAnders Graham Lawrence bboth Brett HanifinSteven Comstock Birthe Smedsrud Ali IrtturkS. Matthew Scott Lingu Tang Gregory Adams
Group D Group E
Troy Dewitt Mats OlsonEmilia Bragadottir Brandon BriggsChristopher Wilderman Theodore EhlertQun Luo Alan WeinbergDane Louvier David Sheehan
6
7
2
4
6
8
10
12
14
70 75 80 85 90 95 00 05
DURATION
Duration of Unemployment 1967.07-2007.04
8
9
10
11
0
10
20
30
40
50
60
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Series: DDURATIONSample 1967:08 2007:04Observations 477
Mean 0.008805Median 0.000000Maximum 1.600000Minimum -2.100000Std. Dev. 0.444630Skewness -0.574108Kurtosis 6.059509
Jarque-Bera 212.2450Probability 0.000000
12
13
14
15
16
17
18
19
20
21
0.0
0.5
1.0
1.5
2.0
70 75 80 85 90 95 00 05
GARCH01
22
2
4
6
8
10
12
14
70 75 80 85 90 95 00 05
DURATION
0.0
0.5
1.0
1.5
2.0
70 75 80 85 90 95 00 05
GARCH01
23
Median Duration of Unemployment in Weeks and Conditional Variance, July '67-April '07
0
2
4
6
8
10
12
14
Date
We
ek
s
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Duration
Conditional Variance
vari
ance
25
26
27
28
29
30
31
Part I. ARCH-M Modeks
In an ARCH-M model, the conditional variance is introduced into the equation for the mean as an explanatory variable.
ARCH-M is often used in financial models
32Net return to an asset model Net return to an asset: y(t)
• y(t) = u(t) + e(t)• where u(t) is is the expected risk premium• e(t) is the asset specific shock
the expected risk premium: u(t)• u(t) = a + b*h(t)• h(t) is the conditional variance
Combining, we obtain:• y(t) = a + b*h(t) +e(t)
33Northern Telecom And Toronto Stock Exchange
Nortel and TSE monthly rates of return on the stock and the market, respectively
Keller and Warrack, 6th ed. Xm 18-06 data file
We used a similar file for GE and S_P_Index01 last Fall in Lab 6 of Econ 240A
34
35Returns Generating Model, Variables Not Net of Risk Free
36
37Diagnostics: Correlogram of the Residuals
38Diagnostics: Correlogram of Residuals Squared
39
40Try Estimating An ARCH-
GARCH Model
41
42Try Adding the Conditional Variance to the Returns Model PROCS: Make GARCH variance series:
GARCH01 series
43Conditional Variance Does Not Explain Nortel Return
44
45OLS ARCH-M
46
Estimate ARCH-M Model
47Estimating Arch-M in Eviews with GARCH
48
49
50Three-Mile Island
51
52
53
54
Event: March 28, 1979
55
56
57Garch01 as a Geometric Lag of GPUnet
Garch01(t) = {b/[1-(1-b)z]} zm gpunet(t) Garch01(t) = (1-b) garch01(t-1) + b zm gpunet
58
59
Part II. Granger Causality
Granger causality is based on the notion of the past causing the present
example: Index of Consumer Sentiment January 1978 - March 2003 and S&P500 total return, monthly January 1970 - March 2003
60Consumer Sentiment and SP 500 Total Return
61
Time Series are Evolutionary
Take logarithms and first difference
62
63
64
Dlncon’s dependence on its past
dlncon(t) = a + b*dlncon(t-1) + c*dlncon(t-2) + d*dlncon(t-3) + resid(t)
65
66Dlncon’s dependence on its past and dlnsp’s past
dlncon(t) = a + b*dlncon(t-1) + c*dlncon(t-2) + d*dlncon(t-3) + e*dlnsp(t-1) + f*dlnsp(t-2) + g* dlnsp(t-3) + resid(t)
67
Do lagged dlnsp terms add to the explained variance?
F3, 292 = {[ssr(eq. 1) - ssr(eq. 2)]/3}/[ssr(eq. 2)/n-7]
F3, 292 = {[0.642038 - 0.575445]/3}/0.575445/292
F3, 292 = 11.26
critical value at 5% level for F(3, infinity) = 2.60
69
Causality goes from dlnsp to dlncon
EVIEWS Granger Causality Test• open dlncon and dlnsp• go to VIEW menu and select Granger Causality• choose the number of lags
70
71Does the causality go the other way, from dlncon to dlnsp? dlnsp(t) = a + b*dlnsp(t-1) + c*dlnsp(t-2) +
d* dlnsp(t-3) + resid(t)
72
73Dlnsp’s dependence on its past and dlncon’s past dlnsp(t) = a + b*dlnsp(t-1) + c*dlnsp(t-2) +
d* dlnsp(t-3) + e*dlncon(t-1) + f*dlncon(t-2) + g*dlncon(t-3) + resid(t)
74
Do lagged dlncon terms add to the explained variance?
F3, 292 = {[ssr(eq. 1) - ssr(eq. 2)]/3}/[ssr(eq. 2)/n-7]
F3, 292 = {[0.609075 - 0.606715]/3}/0.606715/292
F3, 292 = 0.379
critical value at 5% level for F(3, infinity) = 2.60
76
77Granger Causality and Cross-Correlation
One-way causality from dlnsp to dlncon reinforces the results inferred from the cross-correlation function
78
79Part III. Simultaneous Equations
and Identification Lecture 2, Section I Econ 240C Spring
2007 Sometimes in microeconomics it is possible
to identify, for example, supply and demand, if there are exogenous variables that cause the curves to shift, such as weather (rainfall) for supply and income for demand
80
Demand: p = a - b*q +c*y + ep
81
demand
price
quantity
Dependence of price on quantity and vice versa
82
demand
price
quantity
Shift in demand with increased income
83
Supply: q= d + e*p + f*w + eq
84
price
quantity
supply
Dependence of price on quantity and vice versa
85
Simultaneity
There are two relations that show the dependence of price on quantity and vice versa• demand: p = a - b*q +c*y + ep
• supply: q= d + e*p + f*w + eq
86
Endogeneity
Price and quantity are mutually determined by demand and supply, i.e. determined internal to the model, hence the name endogenous variables
income and weather are presumed determined outside the model, hence the name exogenous variables
87
price
quantity
supply
Shift in supply with increased rainfall
88
Identification
Suppose income is increasing but weather is staying the same
89
demand
price
quantity
Shift in demand with increased income, may trace outi.e. identify or reveal the supply curve
supply
90
price
quantity
Shift in demand with increased income, may trace outi.e. identify or reveal the supply curve
supply
91
Identification
Suppose rainfall is increasing but income is staying the same
92
price
quantity
supply
Shift in supply with increased rainfall may trace out, i.e. identify or reveal the demand curve
demand
93
price
quantity
Shift in supply with increased rainfall may trace out, i.e. identify or reveal the demand curve
demand
94
Identification
Suppose both income and weather are changing
95
price
quantity
supply
Shift in supply with increased rainfall and shift in demandwith increased income
demand
96
price
quantity
Shift in supply with increased rainfall and shift in demandwith increased income. You observe price and quantity
97
Identification
All may not be lost, if parameters of interest such as a and b can be determined from the dependence of price on income and weather and the dependence of quantity on income and weather then the demand model can be identified and so can supply
The Reduced Form for p~(y,w)
demand: p = a - b*q +c*y + ep
supply: q= d + e*p + f*w + eq
Substitute expression for q into the demand equation and solve for p
p = a - b*[d + e*p + f*w + eq] +c*y + ep
p = a - b*d - b*e*p - b*f*w - b* eq + c*y + ep
p[1 + b*e] = [a - b*d ] - b*f*w + c*y + [ep - b* eq ]
divide through by [1 + b*e]
The reduced form for q~y,w
demand: p = a - b*q +c*y + ep
supply: q= d + e*p + f*w + eq
Substitute expression for p into the supply equation and solve for q
supply: q= d + e*[a - b*q +c*y + ep] + f*w + eq
q = d + e*a - e*b*q + e*c*y +e* ep + f*w + eq
q[1 + e*b] = [d + e*a] + e*c*y + f*w + [eq + e* ep]
divide through by [1 + e*b]
Working back to the structural parameters
Note: the coefficient on income, y, in the equation for q, divided by the coefficient on income in the equation for p equals e, the slope of the supply equation• e*c/[1+e*b]÷ c/[1+e*b] = e
Note: the coefficient on weather in the equation f for p, divided by the coefficient on weather in the equation for q equals -b, the slope of the demand equation
This process is called identification
From these estimates of e and b we can calculate [1 +b*e] and obtain c from the coefficient on income in the price equation and obtain f from the coefficient on weather in the quantity equation
it is possible to obtain a and d as well
102
Vector Autoregression (VAR)
Simultaneity is also a problem in macro economics and is often complicated by the fact that there are not obvious exogenous variables like income and weather to save the day
As John Muir said, “everything in the universe is connected to everything else”
103VAR One possibility is to take advantage of the
dependence of a macro variable on its own past and the past of other endogenous variables. That is the approach of VAR, similar to the specification of Granger Causality tests
One difficulty is identification, working back from the equations we estimate, unlike the demand and supply example above
We miss our equation specific exogenous variables, income and weather
Primitive VAR
(1)y(t) = w(t) + y(t-1) +
w(t-1) + x(t) + ey(t)
(2) w(t) = y(t) + y(t-1) +
w(t-1) + x(t) + ew(t)
105
Standard VAR
Eliminate dependence of y(t) on contemporaneous w(t) by substituting for w(t) in equation (1) from its expression (RHS) in equation 2
1. y(t) = w(t) + y(t-1) + w(t-1) + x(t) + ey(t)
1’. y(t) = y(t) + y(t-1) + w(t-1) + x(t) + ew(t)] + y(t-1) + w(t-1) + x(t) + ey(t)
1’. y(t) y(t) = [+ y(t-1) + w(t-1) + x(t) + ew(t)] + y(t-1) + w(t-1) + x(t) + ey(t)
Standard VAR (1’) y(t) = (/(1- ) +[ (+
)/(1- )] y(t-1) + [ (+ )/(1- )] w(t-1) + [(+ (1- )] x(t) + (ey(t) + ew(t))/(1- )
in the this standard VAR, y(t) depends only on lagged y(t-1) and w(t-1), called predetermined variables, i.e. determined in the past
Note: the error term in Eq. 1’, (ey(t) + ew(t))/(1- ), depends upon both the pure shock to y, ey(t) , and the pure shock to w, ew
Standard VAR (1’) y(t) = (/(1- ) +[ (+ )/(1-
)] y(t-1) + [ (+ )/(1- )] w(t-1) + [(+ (1- )] x(t) + (ey(t) + ew(t))/(1- )
(2’) w(t) = (/(1- ) +[(+ )/(1- )] y(t-1) + [ (+ )/(1- )] w(t-1) + [(+ (1- )] x(t) + (ey(t) + ew(t))/(1- )
Note: it is not possible to go from the standard VAR to the primitive VAR by taking ratios of estimated parameters in the standard VAR