Transcript Power 15 - UCSB Economics

Econ 240 C
Lecture 15
Outline
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Project II
Forecasting
ARCH-M Models
Granger Causality
Simultaneity
VAR models
2
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 nontechnical 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.
3
The technical details of your findings you can attach as an appendix
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
4
Group A
Group B
Hao Jin
Hamed Farshbaf Dadgur
Hilda Hesjedal
Michael Wiltshire
Bryan Watson
Joshua Greeson
Ching-Chi Huang
Cliff Holt
Amanda Smith
Candace Miller
Peter Fuller
Sean Karafin
Vineet Sharma
Group D
Group E
Chun-Kai Wang
Johan Routh
Chun-Hung Lin
Ryan Degrazier
May Thet Zin
Anastasia Zavodny
Andreas Lindahl
Tsung-Ching Huang
Michael Williams
Juan Shan
Odeliah Greene
Kevin Crider
Group C
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Liu He
Guizi Li
Chien-ju Lin
Lyle Kaplan-Reinig
Matthew Routh
Eduardo Velasquez
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Duration of Unemployment 1967.07-2007.04
7
14
12
10
8
6
4
2
70
75
80
85
90
DURATION
95
00
05
8
9
10
11
60
Series: DDURATION
Sample 1967:08 2007:04
Observations 477
50
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
40
30
20
10
Jarque-Bera
Probability
0
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0.008805
0.000000
1.600000
-2.100000
0.444630
-0.574108
6.059509
212.2450
0.000000
12
13
14
15
16
17
18
19
20
2.0
21
1.5
1.0
0.5
0.0
70
75
80
85
90
GARCH01
95
00
05
14
22
12
10
8
6
4
2
70
75
80
85
90
95
00
05
95
00
05
DURATION
2.0
1.5
1.0
0.5
0.0
70
75
80
85
90
GARCH01
J
Apul-6
Apr-687
Apr-69
Apr-70
Apr-71
Apr-72
Apr-73
Apr-74
Apr-75
Apr-76
Apr-77
Apr-78
Apr-79
Apr-80
Apr-81
Apr-82
Apr-83
Apr-84
Apr-85
Apr-86
Apr-87
Apr-88
Apr-89
Apr-90
Apr-91
Apr-92
Apr-93
Apr-94
Apr-95
Apr-96
Apr-97
Apr-98
Apr-99
Apr-00
Apr-01
Apr-02
Apr-03
Apr-04
Apr-05
Apr-06
r-0
7
Weeks
14
12
Duration
Conditional Variance
10
8
1
6
0.8
4
0.6
2
0.4
0.2
0
0
Date
variance
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Median Duration of Unemployment in Weeks and Conditional Variance, July '67-April '07
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1.8
1.6
1.4
1.2
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http://www.dof.ca.gov/
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Part I. ARCH-M Modeks
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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
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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
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the expected risk premium: u(t)
• u(t) = a + b*h(t)
• h(t) is the conditional variance
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Combining, we obtain:
• y(t) = a + b*h(t) +e(t)
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Northern Telecom And Toronto
Stock Exchange
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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 ARCHGARCH Model
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Try Adding the Conditional
Variance to the Returns Model
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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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Three-Mile Island
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Event: March 28, 1979
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Garch01 as a Geometric Lag 57of
GPUnet
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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
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Granger causality is based on the notion of
the past causing the present
example: Index of Consumer Sentiment
January 1978 - March 2003 and S&P500
total return, monthly January 1970 - March
2003
Consumer Sentiment and SP 500 Total Return
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Time Series are Evolutionary
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Take logarithms and first difference
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Dlncon’s dependence on its past
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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
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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?
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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
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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?
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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
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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?
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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
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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
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Lecture 2, Section I Econ 240C Spring 2007
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
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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
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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
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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 supply 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
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Suppose rainfall is increasing but income is
staying the same
Shift in supply with increased rainfall may trace out, 92
i.e. identify or reveal the demand curve
price
demand
supply
quantity
Shift in supply with increased rainfall may trace out, 93
i.e. identify or reveal the demand curve
price
demand
quantity
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Identification
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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 quantity
price
quantity
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Identification
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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)
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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
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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
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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
• e*c/[1+e*b]÷ c/[1+e*b] = e
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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
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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)
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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
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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) =   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
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(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
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(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