Econ 240 C Lecture 16

2 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

3 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)

4 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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6 Returns Generating Model, Variables Not Net of Risk Free

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8 Diagnostics: Correlogram of the Residuals

9 Diagnostics: Correlogram of Residuals Squared

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11 Try Estimating An ARCH- GARCH Model

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

14 Conditional Variance Does Not Explain Nortel Return

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16 OLS ARCH-M

17 Estimate ARCH-M Model

18 Estimating Arch-M in Eviews with GARCH

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

20 Consumer Sentiment and SP 500 Total Return

21 Time Series are Evolutionary w Take logarithms and first difference

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24 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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26 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

29 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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31 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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33 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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37 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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39 Part III. Simultaneous Equations and Identification w Lecture 2, Section I Econ 240C Spring 2004 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

40 w Demand: p = a - b*q +c*y + e p

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

42 demand price quantity Shift in demand with increased income

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

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

45 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

46 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

47 price quantity supply Shift in supply with increased rainfall

48 Identification w Suppose income is increasing but weather is staying the same

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

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

51 Identification w Suppose rainfall is increasing but income is staying the same

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

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

54 Identification w Suppose both income and weather are changing

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

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

57 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 for 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

62 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”

63 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

65 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