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Econ 240 C
Lecture 16
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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
Net 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)
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Northern 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
240C
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Returns Generating Model, Variables Not Net of Risk Free
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Diagnostics: Correlogram of the Residuals
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Diagnostics: Correlogram of Residuals Squared
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Try Estimating An ARCHGARCH Model
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Try Adding the Conditional
Variance to the Returns Model
PROCS: Make GARCH variance series:
GARCH01 series
Conditional Variance Does Not Explain Nortel Return
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OLS ARCH-M
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Estimate ARCH-M Model
Estimating Arch-M in Eviews with GARCH
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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
Consumer 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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Dlncon’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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Does 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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Dlnsp’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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Granger Causality and CrossCorrelation
One-way causality from dlnsp to dlncon
reinforces the results inferred from the
cross-correlation function
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Part III. Simultaneous Equations
and Identification
Lecture 2, Section I Econ 240C Spring 2003
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
Dependence of price on quantity and vice versa
price
demand
quantity
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Shift in demand with increased income
price
demand
quantity
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Supply: q= d + e*p + f*w + eq
Dependence of price on quantity and vice versa
price
supply
quantity
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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
Shift in supply with increased rainfall
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price
supply
quantity
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Identification
Suppose income is increasing but weather is
staying the same
Shift in demand with increased income, may trace out
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i.e. identify or reveal the demand curve
price
supply
demand
quantity
Shift in demand with increased income, may trace out
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i.e. identify or reveal the supply curve
price
supply
quantity
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Identification
Suppose rainfall is increasing but income is
staying the same
Shift in supply with increased rainfall may trace out, 52
i.e. identify or reveal the demand curve
price
demand
supply
quantity
Shift in supply with increased rainfall may trace out, 53
i.e. identify or reveal the demand curve
price
demand
quantity
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Identification
Suppose both income and weather are
changing
Shift in supply with increased rainfall and shift in demand
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with increased income
price
demand
supply
quantity
Shift in supply with increased rainfall and shift in demand
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with increased income. You observe price and income
price
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]
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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
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
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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”
VAR
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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(t1) + 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
