Econ 240 C

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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

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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 bh(t)
• h(t) is the conditional variance
• Combining, we obtain
• y(t) a bh(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 240A

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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 ARCH-GARCH Model
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Try Adding the Conditional Variance to the
Returns Model

• PROCS Make GARCH variance series GARCH01 series

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Conditional Variance Does Not Explain Nortel
Return
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OLS ARCH-M
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Estimate ARCH-M Model
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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 SP500 total
  return, montly January 1970 - March 2003

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Consumer Sentiment and SP 500 Total Return
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Time Series are Evolutionary

• Take logarithms and first difference

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Dlncons dependence on its past

• dlncon(t) a bdlncon(t-1) cdlncon(t-2)
  ddlncon(t-3) resid(t)

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Dlncons dependence on its past and dlnsps past

• dlncon(t) a bdlncon(t-1) cdlncon(t-2)
  ddlncon(t-3) edlnsp(t-1)
  fdlnsp(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 bdlnsp(t-1) cdlnsp(t-2) d
  dlnsp(t-3) resid(t)

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Dlnsps dependence on its past and dlncons past

• dlnsp(t) a bdlnsp(t-1) cdlnsp(t-2) d
  dlnsp(t-3) edlncon(t-1)
  fdlncon(t-2) gdlncon(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 Cross-Correlation

• 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 2004
• 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 - bq cy ep

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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 ep fw eq

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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 - bq cy ep
• supply q d ep fw 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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Shift in supply with increased rainfall
price
supply
quantity
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Identification

• Suppose income is increasing but weather is
  staying the same

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

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

• Suppose both income and weather are changing

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

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The Reduced Form for p(y,w)

• demand p a - bq cy ep
• supply q d ep fw eq
• Substitute expression for q into the demand
  equation and solve for p
• p a - bd ep fw eq cy ep
• p a - bd - bep - bfw - b eq cy ep
• p1 be a - bd - bfw cy ep - b
  eq
• divide through by 1 be

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The reduced form for qy,w

• demand p a - bq cy ep
• supply q d ep fw eq
• Substitute expression for p into the supply
  equation and solve for q
• supply q d ea - bq cy ep fw eq
• q d ea - ebq ecy e ep fw eq
• q1 eb d ea ecy fw eq e
  ep
• divide through by 1 eb

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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
• ec/1eb c/1eb e
• 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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This process is called identification

• From these estimates of e and b we can calculate
  1 be 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

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VAR

• 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

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Primitive VAR
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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

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• 1. y(t) a1 b1 w(t) g11 y(t-1) g12 w(t-1)
  d1 x(t) ey (t)
• 1. y(t) a1 b1 a2 b2 y(t) g21 y(t-1)
  g22 w(t-1) d2 x(t) ew (t) g11 y(t-1) g12
  w(t-1) d1 x(t) ey (t)
• 1. y(t) - b1b2 y(t) a1 b1 a2 b1g21
  y(t-1) b1g22 w(t-1) b1d2 x(t) b1ew (t)
  g11 y(t-1) g12 w(t-1) d1 x(t) ey (t)

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Standard VAR

• (1) y(t) (a1 b1 a2)/(1- b1 b2) (g11 b1
  g21)/(1- b1 b2) y(t-1) (g12 b1 g22)/(1- b1
  b2) w(t-1) (d1 b1 d2 )/(1- b1 b2) x(t)
  (ey (t) b1 ew (t))/(1- b1 b2)
• 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) b1 ew
  (t))/(1- b1 b2), depends upon both the pure shock
  to y, ey (t) , and the pure shock to w, ew

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Standard VAR

• (1) y(t) (a1 b1 a2)/(1- b1 b2) (g11 b1
  g21)/(1- b1 b2) y(t-1) (g12 b1 g22)/(1- b1
  b2) w(t-1) (d1 b1 d2 )/(1- b1 b2) x(t)
  (ey (t) b1 ew (t))/(1- b1 b2)
• (2) w(t) (b2 a1 a2)/(1- b1 b2) (b2 g11
  g21)/(1- b1 b2) y(t-1) (b2 g12 g22)/(1-
  b1 b2) w(t-1) (b2 d1 d2 )/(1- b1 b2) x(t)
  (b2 ey (t) ew (t))/(1- b1 b2)
• 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