econ 240 c lecture 16. 2 outline w project i w arch-m models w granger causality w simultaneity w...
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Econ 240 C
Lecture 16
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Outline Project I ARCH-M Models Granger Causality Simultaneity VAR models
3Project I Models for dduration Models for dlnduration Seasonality Conditional heteroskedasticity
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Models for ∆duration
Ufook Sahillioghlu• Ar(1) ar(2) ar(4) ar(5) ar(6) ma(7) ma(24) ma(36)
Tom Bruister• Ar(1) ar(2) ar(24) ma(1) ma(4)
Jesse Smith• Ar(1) ar(4) ar(24) ar(36)
5Models for ∆lnduration Jonathan Hester
• Ar(1) ma(1) ma(2) ma(3) Ashley Hedberg
• Ar(1) ar(2) ma(1) ma(2) Jonathan Liu
• Ar(1) ar(2) ar(4) ar(5) ar(6) ma(7) ma(24) ma(36) Yana Ten
• Ma(1) ma(4) ar(24) ar(36) Jeff Ahlvin
• Ma(1) ma(2) ma(3) sma(24)
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Conditional Variance, h
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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
14Net 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)
15Northern 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
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17Returns Generating Model, Variables Not Net of Risk Free
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19Diagnostics: Correlogram of the Residuals
20Diagnostics: Correlogram of Residuals Squared
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22Try Estimating An ARCH-
GARCH Model
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24Try Adding the Conditional Variance to the Returns Model PROCS: Make GARCH variance series:
GARCH01 series
25Conditional Variance Does Not Explain Nortel Return
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27OLS ARCH-M
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Estimate ARCH-M Model
29Estimating Arch-M in Eviews with GARCH
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32Three-Mile Island
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Event: March 28, 1979
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39Garch01 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
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Part II. Granger Causality
Granger causality is based on the notion of the past causing the present
example: Lab six, Index of Consumer Sentiment January 1978 - March 2003 and S&P500 total return, montly January 1970 - March 2003
42Consumer Sentiment and SP 500 Total Return
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Time Series are Evolutionary
Take logarithms and first difference
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Dlncon’s dependence on its past
dlncon(t) = a + b*dlncon(t-1) + c*dlncon(t-2) + d*dlncon(t-3) + resid(t)
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48Dlncon’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)
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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
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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
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53Does 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)
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55Dlnsp’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)
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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
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59Granger Causality and Cross-Correlation
One-way causality from dlnsp to dlncon reinforces the results inferred from the cross-correlation function
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61Part III. Simultaneous Equations
and Identification Lecture 2, Section I Econ 240C Spring
2005 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
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Demand: p = a - b*q +c*y + ep
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demand
price
quantity
Dependence of price on quantity and vice versa
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demand
price
quantity
Shift in demand with increased income
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Supply: q= d + e*p + f*w + eq
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price
quantity
supply
Dependence of price on quantity and vice versa
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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
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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
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price
quantity
supply
Shift in supply with increased rainfall
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Identification
Suppose income is increasing but weather is staying the same
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demand
price
quantity
Shift in demand with increased income, may trace outi.e. identify or reveal the demand curve
supply
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price
quantity
Shift in demand with increased income, may trace outi.e. identify or reveal the supply curve
supply
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Identification
Suppose rainfall is increasing but income is staying the same
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price
quantity
supply
Shift in supply with increased rainfall may trace out, i.e. identify or reveal the demand curve
demand
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price
quantity
Shift in supply with increased rainfall may trace out, i.e. identify or reveal the demand curve
demand
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Identification
Suppose both income and weather are changing
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price
quantity
supply
Shift in supply with increased rainfall and shift in demandwith increased income
demand
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price
quantity
Shift in supply with increased rainfall and shift in demandwith increased income. You observe price and quantity
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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
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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”
85VAR 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)
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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