Econ 240 C Lecture 15

2 Outline w Project II w Forecasting w ARCH-M Models w Granger Causality w Simultaneity w VAR models

3 I. Work in Groups II. 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 Appendix 1. Table of Contents 2. Spreadsheet of data used and sources or, if extensive, a subsample of the data 3. Describe the analytical time series techniques you are using 4. Show descriptive statistics and histograms for the variables in the study 5. 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

5 Group AGroup BGroup C Hao JinHamed Farshbaf DadgurLiu He Hilda HesjedalMichael WiltshireGuizi Li Bryan WatsonJoshua GreesonChien-ju Lin Ching-Chi HuangCliff HoltLyle Kaplan-Reinig Amanda SmithCandace MillerMatthew Routh Peter FullerSean KarafinEduardo Velasquez Vineet Sharma Group DGroup E Chun-Kai WangAndreas Lindahl Johan RouthTsung-Ching Huang Chun-Hung Lin Michael Williams Ryan DegrazierJuan Shan May Thet ZinOdeliah Greene Anastasia ZavodnyKevin Crider

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31 Part I. ARCH-M Modeks w In an ARCH-M model, the conditional variance is introduced into the equation for the mean as an explanatory variable. w ARCH-M is often used in financial models

32 Net return to an asset model w 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 w the expected risk premium: u(t) u(t) = a + b*h(t) h(t) is the conditional variance w Combining, we obtain: y(t) = a + b*h(t) +e(t)

33 Northern Telecom And Toronto Stock Exchange w Nortel and TSE monthly rates of return on the stock and the market, respectively w Keller and Warrack, 6th ed. Xm data file w We used a similar file for GE and S_P_Index01 last Fall in Lab 6 of Econ 240A

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35 Returns Generating Model, Variables Not Net of Risk Free

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37 Diagnostics: Correlogram of the Residuals

38 Diagnostics: Correlogram of Residuals Squared

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40 Try Estimating An ARCH- GARCH Model

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42 Try Adding the Conditional Variance to the Returns Model w PROCS: Make GARCH variance series: GARCH01 series

43 Conditional Variance Does Not Explain Nortel Return

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45 OLS ARCH-M

46 Estimate ARCH-M Model

47 Estimating Arch-M in Eviews with GARCH

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50 Three-Mile Island

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54 Event: March 28, 1979

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57 Garch01 as a Geometric Lag of GPUnet w Garch01(t) = {b/[1-(1-b)z]} z m gpunet(t) w Garch01(t) = (1-b) garch01(t-1) + b z m gpunet

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59 Part II. Granger Causality w Granger causality is based on the notion of the past causing the present w example: Index of Consumer Sentiment January March 2003 and S&P500 total return, monthly January March 2003

60 Consumer Sentiment and SP 500 Total Return

61 Time Series are Evolutionary w Take logarithms and first difference

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64 Dlncon’s dependence on its past w dlncon(t) = a + b*dlncon(t-1) + c*dlncon(t-2) + d*dlncon(t-3) + resid(t)

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66 Dlncon’s dependence on its past and dlnsp’s past w 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)

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Do lagged dlnsp terms add to the explained variance? w F 3, 292 = {[ssr(eq. 1) - ssr(eq. 2)]/3}/[ssr(eq. 2)/n-7] w F 3, 292 = {[ ]/3}/ /292 w F 3, 292 = w critical value at 5% level for F(3, infinity) = 2.60

69 Causality goes from dlnsp to dlncon w EVIEWS Granger Causality Test open dlncon and dlnsp go to VIEW menu and select Granger Causality choose the number of lags

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71 Does the causality go the other way, from dlncon to dlnsp? w dlnsp(t) = a + b*dlnsp(t-1) + c*dlnsp(t-2) + d* dlnsp(t-3) + resid(t)

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73 Dlnsp’s dependence on its past and dlncon’s past w 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)

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Do lagged dlncon terms add to the explained variance? w F 3, 292 = {[ssr(eq. 1) - ssr(eq. 2)]/3}/[ssr(eq. 2)/n-7] w F 3, 292 = {[ ]/3}/ /292 w F 3, 292 = w critical value at 5% level for F(3, infinity) = 2.60

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77 Granger Causality and Cross- Correlation w One-way causality from dlnsp to dlncon reinforces the results inferred from the cross-correlation function

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79 Part III. Simultaneous Equations and Identification w Lecture 2, Section I Econ 240C Spring 2007 w 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 w Demand: p = a - b*q +c*y + e p

81 demand price quantity Dependence of price on quantity and vice versa

82 demand price quantity Shift in demand with increased income

83 w Supply: q= d + e*p + f*w + e q

84 price quantity supply Dependence of price on quantity and vice versa

85 Simultaneity w There are two relations that show the dependence of price on quantity and vice versa demand: p = a - b*q +c*y + e p supply: q= d + e*p + f*w + e q

86 Endogeneity w Price and quantity are mutually determined by demand and supply, i.e. determined internal to the model, hence the name endogenous variables w 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 w Suppose income is increasing but weather is staying the same

89 demand price quantity Shift in demand with increased income, may trace out i.e. identify or reveal the supply curve supply

90 price quantity Shift in demand with increased income, may trace out i.e. identify or reveal the supply curve supply

91 Identification w 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 w Suppose both income and weather are changing

95 price quantity supply Shift in supply with increased rainfall and shift in demand with increased income demand

96 price quantity Shift in supply with increased rainfall and shift in demand with increased income. You observe price and quantity

97 Identification w 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) w demand: p = a - b*q +c*y + e p w supply: q= d + e*p + f*w + e q w Substitute expression for q into the demand equation and solve for p w p = a - b*[d + e*p + f*w + e q ] +c*y + e p w p = a - b*d - b*e*p - b*f*w - b* e q + c*y + e p w p[1 + b*e] = [a - b*d ] - b*f*w + c*y + [e p - b* e q ] w divide through by [1 + b*e]

The reduced form for q~y,w w demand: p = a - b*q +c*y + e p w supply: q= d + e*p + f*w + e q w Substitute expression for p into the supply equation and solve for q w supply: q= d + e*[a - b*q +c*y + e p ] + f*w + e q w q = d + e*a - e*b*q + e*c*y +e* e p + f*w + e q w q[1 + e*b] = [d + e*a] + e*c*y + f*w + [e q + e* e p ] w divide through by [1 + e*b]

Working back to the structural parameters w 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 w 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 w 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 w it is possible to obtain a and d as well

102 Vector Autoregression (VAR) w 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 w As John Muir said, “everything in the universe is connected to everything else”

103 VAR w 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 w One difficulty is identification, working back from the equations we estimate, unlike the demand and supply example above w We miss our equation specific exogenous variables, income and weather

Primitive VAR

105 Standard VAR w 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) + e y  (t)  1’.y(t) =          y(t) +    y(t-1) +   w(t-1) +    x(t) + e w  (t)] +    y(t-1) +    w(t-1) +    x(t) + e y  (t)  1’.y(t)      y(t) = [        +      y(t-1) +      w(t-1) +      x(t) +   e w  (t)] +    y(t- 1) +    w(t-1) +    x(t) + e y  (t)

Standard VAR  (1’) y(t) = (        /(1-     ) +[ (    +     )/(1-     )] y(t-1) + [ (    +     )/(1-     )] w(t-1) + [(    +      (1-     )] x(t) + (e y  (t) +   e w  (t))/(1-     ) w 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’, (e y  (t) +   e w (t))/(1-     ), depends upon both the pure shock to y, e y  (t), and the pure shock to w, e w

Standard VAR  (1’) y(t) = (        /(1-     ) +[ (    +     )/(1-     )] y(t-1) + [ (    +     )/(1-     )] w(t-1) + [(    +      (1-     )] x(t) + (e y  (t) +   e w  (t))/(1-     )  (2’) w(t) = (        /(1-     ) +[(      +   )/(1-     )] y(t-1) + [ (      +   )/(1-     )] w(t-1) + [(      +    (1-     )] x(t) + (    e y  (t) + e w  (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