Transcript 10. Dynamic Models, Time Series, Panels and Nonstationary Data

Part 10: Time Series Applications [ 1/64]
Econometric Analysis of Panel Data
William Greene
Department of Economics
Stern School of Business
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Dear professor Greene,
I have a plan to run a (endogenous or exogenous) switching regression model
for a panel data set. To my knowledge, there is no routine for this in other
software, and I am not so good at coding a program. Fortunately, I am advised
that LIMDEP has a built in function (or routine) for the panel switch model.
Part 10: Time Series Applications [ 5/64]
Endogenous Switching (ca.1980)
Regime 0: yi  xi 0 0  i 0
Regime 1: yi  xi1 1  i1
Regime Switch: d* = z i   u , d = 1[d* > 0]
Regime 0 governs if d = 0, Probability = 1- (z i  )
Regime 1 governs if d = 1, Probability =  (z i  )
Not identified. Regimes
do not coexist.
 0   02

 i 0 
  

 
Endogenous Switching:  i 0  ~ N  0  ,  ?
12

 
 0       1  
1 1
 i0 

   0 0
This is a latent class model with different processes in the two classes.
There is correlation between the unobservables that govern the class
determination and the unobservables in the two regime equations.
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Modeling an Economic
Time Series
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Observed y0, y1, …, yt,…
What is the “sample”
Random sampling?
The “observation window”
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Estimators
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Functions of sums of observations
Law of large numbers?
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Nonindependent observations
What does “increasing sample size” mean?
Asymptotic properties? (There are no finite
sample properties.)
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Interpreting a Time Series
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Time domain: A “process”
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Frequency domain: A sum of terms
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y(t) = ax(t) + by(t-1) + …
Regression like approach/interpretation

y(t) = j  j Cos ( j t )  (t )
Contribution of different frequencies to the observed
series.
(“High frequency data and financial econometrics
– “frequency” is used slightly differently here.)
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For example,…
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In parts…
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Studying the Frequency Domain
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Cannot identify the number of terms
Cannot identify frequencies from the time series
Deconstructing the variance, autocovariances
and autocorrelations
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Contributions at different frequencies
Apparent large weights at different frequencies
Using Fourier transforms of the data
Does this provide “new” information about the
series?
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Autocorrelation in Regression
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Yt = b’xt + εt
Cov(εt, εt-1) ≠ 0
Ex. RealConst = a + bRealIncome + εt U.S. Data, quarterly, 1950-2000
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Autocorrelation
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How does it arise?
What does it mean?
Modeling approaches
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Classical – direct: corrective
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Estimation that accounts for autocorrelation
Inference in the presence of autocorrelation
Contemporary – structural
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Model the source
Incorporate the time series aspect in the model
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Stationary Time Series
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zt = b1yt-1 + b2yt-2 + … + bPyt-P + et
Autocovariance: γk = Cov[yt,yt-k]
Autocorrelation: k = γk / γ0
Stationary series: γk depends only on k, not on t
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
Weak stationarity: E[yt] is not a function of t, E[yt * yt-s] is not a
function of t or s, only of |t-s|
Strong stationarity: The joint distribution of [yt,yt-1,…,yt-s] for any
window of length s periods, is not a function of t or s.
A condition for weak stationarity: The smallest root of the
characteristic polynomial: 1 - b1z1 - b2z2 - … - bPzP = 0, is greater
than one.
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The unit circle
Complex roots
Example: yt = yt-1 + ee, 1 - z = 0 has root z = 1/ ,
| z | > 1 => |  | < 1.
Part 10: Time Series Applications [ 17/64]
Stationary vs. Nonstationary Series
Part 10: Time Series Applications [ 18/64]
The Lag Operator
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Lc = c when c is a constant
Lxt = xt-1
L2 xt = xt-2
LPxt + LQxt = xt-P + xt-Q
Polynomials in L: yt = B(L)yt + et
A(L) yt = et
Invertibility: yt = [A(L)]-1 et
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Inverting a Stationary Series
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yt= yt-1 + et  (1- L)yt = et
yt = [1- L]-1 et = et + et-1 + 2et-2 + …
1
 1  (L)  (L) 2  (L)3  ...
1  L
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Stationary series can be inverted
Autoregressive vs. moving average form of series
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Regression with Autocorrelation
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yt
= xt’b + et, et = et-1 + ut
(1- L)et = ut  et = (1- L)-1ut
 E[et]
= E[ (1- L)-1ut] = (1- L)-1E[ut] = 0
 Var[et] = (1- L)-2Var[ut] = 1+ 2u2 + …
= u2/(1- 2)
 Cov[et,et-1] = Cov[et-1 + ut, et-1] =
=Cov[et-1,et-1]+Cov[ut,et-1]
=  u2/(1- 2)
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OLS vs. GLS
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OLS
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Unbiased?
Consistent: (Except in the presence of a lagged
dependent variable)
Inefficient
GLS
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Consistent and efficient
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+----------------------------------------------------+
| Ordinary
least squares regression
|
| LHS=REALCONS Mean
=
2999.436
|
| Autocorrel
Durbin-Watson Stat. =
.0920480
|
|
Rho = cor[e,e(-1)]
=
.9539760
|
+----------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant
-80.3547488
14.3058515
-5.617
.0000
REALDPI
.92168567
.00387175
238.054
.0000
3341.47598
| Robust VC
Newey-West, Periods =
10
|
Constant
-80.3547488
41.7239214
-1.926
.0555
REALDPI
.92168567
.01503516
61.302
.0000
3341.47598
+---------------------------------------------+
| AR(1) Model:
e(t) = rho * e(t-1) + u(t) |
| Final value of Rho
=
.998782 |
| Iter= 6, SS= 118367.007, Log-L=-941.371914 |
| Durbin-Watson:
e(t) =
.002436 |
| Std. Deviation: e(t) =
490.567910 |
| Std. Deviation: u(t) =
24.206926 |
| Durbin-Watson:
u(t) =
1.994957 |
| Autocorrelation: u(t) =
.002521 |
| N[0,1] used for significance levels
|
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant
1019.32680
411.177156
2.479
.0132
REALDPI
.67342731
.03972593
16.952
.0000
3341.47598
RHO
.99878181
.00346332
288.389
.0000
Part 10: Time Series Applications [ 23/64]
Detecting Autocorrelation

Use residuals
Tt2 (e t et 1 )2
 2(1  r )
T
2
 Durbin-Watson d=
t 1et
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Assumes normally distributed disturbances strictly
exogenous regressors
Variable addition (Godfrey)
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yt = ’xt + εt-1 + ut
Use regression residuals et and test  = 0
Assumes consistency of b.
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A Unit Root?
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How to test for  = 1?
By construction: εt – εt-1 = ( - 1)εt-1 + ut
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Test for γ = ( - 1) = 0 using regression?
Variance goes to 0 faster than 1/T. Need a new
table; can’t use standard t tables.
Dickey – Fuller tests
Unit roots in economic data. (Are there?)
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Nonstationary series
Implications for conventional analysis
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Reinterpreting Autocorrelation
Regression form
yt   ' xt  t , t  t 1  ut
Error Correction Form
yt  yt 1   '( xt  xt 1 )  ( yt 1   ' xt 1 )  ut , (    1)
 ' xt  the equilibrium
The model describes adjustment of y t to equilibrium when
x t changes.
Part 10: Time Series Applications [ 26/64]
Integrated Processes
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Integration of order (P) when the P’th
differenced series is stationary
Stationary series are I(0)
Trending series are often I(1). Then yt – yt-1 =
yt is I(0). [Most macroeconomic data
series.]
Accelerating series might be I(2). Then
(yt – yt-1)- (yt – yt-1) = 2yt is I(0) [Money stock
in hyperinflationary economies. Difficult to find
many applications in economics]
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Cointegration: Real DPI and Real
Consumption
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Cointegration – Divergent Series?
Part 10: Time Series Applications [ 29/64]
Cointegration
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X(t) and y(t) are obviously I(1)
Looks like any linear combination of x(t) and y(t) will
also be I(1)
Does a model y(t) = bx(t) + u(u) where u(t) is I(0)
make any sense? How can u(t) be I(0)?
In fact, there is a linear combination, [1,-] that is I(0).
y(t) = .1*t + noise, x(t) = .2*t + noise
y(t) and x(t) have a common trend
y(t) and x(t) are cointegrated.
Part 10: Time Series Applications [ 30/64]
Cointegration and I(0) Residuals
Part 10: Time Series Applications [ 31/64]
Cross Country Growth Convergence
Solow - Swan Growth Model

Yi,t  K i,t
(A i,tL i,t )1
A i,t  index of technology
K i,t  capital stock = Ii,t-1 + (1-)K i,t-1
Ii,t 1  s i Yi,t 1  Investment  Savings
L i,t  labor  (1  ni )L i,t 1
Equilibrium Model for Steady State Income
logYi,t  i  i t  i log Yi,t 1  i,t
i  (1-i )gi , gi=technological growth rate
i  "convergence" parameter; 1-i = rate of convergence to steady state
Cross country comparisons of convergence rates are the focus of study.
Part 10: Time Series Applications [ 32/64]
Heterogeneous Dynamic Model
logYi,t  i   i log Yi,t 1  i x it  i,t
Long run effect of interest is i 
i
1  i

1
(1) "Fixed Effects:" Separate regressions, then average results.
Average (over countries) effect:  or
(2) Country means (over time) - can be manipulated to produce
consistent estimators of desired parameters
(3) Time series of means across countries (does not work at all)
(4) Pooled - no way to obtain consistent estimates.
(5) Mixed Fixed (Weinhold - Hsiao): Build separate i into the
equation with "fixed effects," treat i    ui as random.
Part 10: Time Series Applications [ 33/64]
“Fixed Effects” Approach
logYi,t  i  i log Yi,t 1  i x it  i,t
i 
i

, =
1  i
1
ˆi , ˆ
(1) Separate regressions; 
i
ˆi , 
i
1 N ˆ
1 Nˆ
ˆ
(2) Average estimates =   i1
 i1i or
ˆ
N
1  i N
ˆ=
Function of averages: 
(1 / N)Ni1ˆ
i
ˆi
1  (1 / N)Ni1
In each case, each term i has variance O(1/Ti )
Each average has variance O(1/N) 

N
i=1
(1/N)O(1/Ti )
Expect consistency of estimates of long run effects.
Part 10: Time Series Applications [ 34/64]
Country Means
logYi,t  i  i log Yi,t 1  i x it  i,t
i 
i

, =
1  i
1
logYi  i  i log Yi,1  i x i  i
i contains log Yi,1
Estimates are inconsistent. But,
 tT1 log y it
logYi 
,
Ti
log Yi,1 
 tT1 log y i,t 1
Ti
 logYi 
y i,T  y i,0
Ti
 logYi   T (y) / Ti
logYi  i  i log Yi,1  i x i  i  i  i (logYi   T (y) / Ti )  i x i  i

i
i
i
i

xi 
 T (y) / Ti 
1  i 1  i
1  i
1  i
Part 10: Time Series Applications [ 35/64]
Country Means (cont.)
logYi  i   i log Yi,1  i x i  i  i   i (logYi   T (y) / Ti )  i x i  i

i
i
i
i

xi 
 T (y) / Ti 
1  i 1  i
1  i
1  i
(1) Let = i0  i /(1  i ), expect i0  0 to be random
i =i /(1  i ) and i   to be random
(2) Level variable x i should be uncorrelated with change (logYi   T (y) / Ti ).
Regression of logYi on 1, x i should give consistent estimates of 0 and .
Part 10: Time Series Applications [ 36/64]
Time Series
log y t  (1 / N)Ni1 log y i,t
=  + (1/N)Ni1i y i,t 1  (1/N)Ni1i x i,t  (1/N)Ni1i,t
Use =(1/N)Ni1i so i    (i  ) then
(1/N)Ni1i log y i,t 1   log y 1,t  (1/N)Ni1 (i  ) log y i,t 1
Likewise for (1/N)Ni1i x i,t
log y t     log y 1,t  x t  ( t  (1/N)Ni1 (i  ) log y i,t 1 ...)
Disturbance is correlated with the regressor. There is
no way out, and by construction no instrumental variable
that could be correlated with the regressor and not the
disturbance.
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Pooling
Essentially the same as the time series case.
OLS or GLS are inconsistent
There could be no instrument that would work
(by construction)
Part 10: Time Series Applications [ 38/64]
A Mixed/Fixed Approach
log y i,t  i  Ni1idi,t log y i,t 1  i x i,t  i,t
di,t = country specific dummy variable.
Treat i and i as random, i is a 'fixed effect.'
This model can be fit consistently by OLS and
efficiently by GLS.
Part 10: Time Series Applications [ 39/64]
A Mixed Fixed Model Estimator
log y i,t  i  Ni1idi,t log y i,t 1  x i,t  (w i x i,t  i,t )
i    wi
2
Heteroscedastic : Var[w i x i,t  i,t ]=2w x i,t
 2
Use two step least squares.
(1) Linear regression of logy i,t on dummy variables, dummy
variables times logy i,t-1 and x i,t .
(2) Regress squares of OLS residuals on x i,t2 and 1 to
estimate 2w and 2 .
(3) Return to (1) but now use weighted least squares.
Part 10: Time Series Applications [ 40/64]
Nair-Reichert and Weinhold on Growth
Weinhold (1996) and Nair–Reichert and Weinhold (2001) analyzed growth and
development in a panel of 24 developing countries observed for 25 years, 1971–1995. The
model they employed was a variant of the mixed-fixed model proposed by Hsiao (1986,
2003). In their specification,
GGDPi,t = αi + γi dit GGDPi,t-1 + β1i GGDIi,t-1 + β2i GFDIi,t-1
+ β3i GEXPi,t-1 + β4 INFLi,t-1 + εi,t
GGDP = Growth rate of gross domestic product,
GGDI = Growth rate of gross domestic investment,
GFDI = Growth rate of foreign direct investment (inflows),
GEXP = Growth rate of exports of goods and services,
INFL = Inflation rate.
The constant terms and coefficients on the lagged dependent variable are country specific.
The remaining coefficients are treated as random, normally distributed, with means βk and
unrestricted variances. They are modeled as uncorrelated. The model was estimated using
a modification of the Hildreth–Houck–Swamy method
Part 10: Time Series Applications [ 41/64]
Analysis of Macroeconomic Data
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Integrated series
The problem with regressions involving nonstationary
series
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Solutions to the “problem”
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Spurious regressions
Unit roots and misleading relationships
Random walks and first differencing
Removing common trends
Cointegration: Formal solutions to regression models
involving nonstationary data
Extending these results to panels
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Large T and small T cases.
Parameter heterogeneity across countries
Part 10: Time Series Applications [ 42/64]
Nonstationary Data
Part 10: Time Series Applications [ 43/64]
Integrated Series
Part 10: Time Series Applications [ 44/64]
Stationary Data
Part 10: Time Series Applications [ 45/64]
Unit Root Tests
y t   0 y t 1     t  Ll1 l y t l   t
Different restrictions on parameters produce the model.
The parameter of interest is  0 .
Augmented Dickey-Fuller Tests:
Unit root test H0 :  < 1 vs. H1 :   1.
KPSS Tests:
Null hypothesis is stationarity. Alternative is broadly
defined as nonstationary.
Part 10: Time Series Applications [ 46/64]
KPSS Test-1
Part 10: Time Series Applications [ 47/64]
KPSS Test-2
Part 10: Time Series Applications [ 48/64]
Cointegrated Variables?
Part 10: Time Series Applications [ 49/64]
Cointegrating Relationships

Implications:

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Long run vs. short run relationships
Problems of spurious regressions (as
usual)
Problem for existing empirical
studies: Regressions involving
variables of different integration.
E.g., regressions of flows on stocks
Part 10: Time Series Applications [ 50/64]
Money demand example
Part 10: Time Series Applications [ 51/64]
Panel Unit Root Tests
Part 10: Time Series Applications [ 52/64]
Implications
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Separate analyses by country
How to combine data and test statistics
Cointegrating relationships across countries
Part 10: Time Series Applications [ 53/64]
Purchasing Power Parity
Cross country purchasing power parity hypothesis:
Ei,t  i  iPi,t  i,t
Ei,t  log of exchange rate country i with U.S.
Pi,t  log of aggregate consumer expenditure price ratio
Hypothesis of PPP is i  1.
Data on numerous countries (large N) and many periods (large T)
Standard simple regressions based on SUR model?
(See Pedroni, P., "Purchasing Power Parity Tests in Cointegrated
Panels," ReStat, Nov, 2001, p. 727 and related cited papers.)
Part 10: Time Series Applications [ 54/64]
Application
“Some international evidence on price determination: a non-stationary
panel Approach,” Paul Ashworth, Joseph P. Byrne, Economic Modelling,
20, 2003, p. 809-838.
80 quarters, 13 OECD countries
log pi,t = β0 + β1log(unit labor costi,t)
+ β2 log(world price,t)
+ β3 log(intermediate goods pricei,t)
+ β4 (log-output gapi,t) + εi,t
Various tests for unit roots and cointegration
Part 10: Time Series Applications [ 55/64]
Vector Autoregression
The vector autoregression (VAR) model is one of the most successful, flexible,
and easy to use models for the analysis of multivariate time series. It is
a natural extension of the univariate autoregressive model to dynamic multivariate
time series. The VAR model has proven to be especially useful for
describing the dynamic behavior of economic and financial time series and
for forecasting. It often provides superior forecasts to those from univariate
time series models and elaborate theory-based simultaneous equations
models. Forecasts from VAR models are quite flexible because they can be
made conditional on the potential future paths of specified variables in the
model.
In addition to data description and forecasting, the VAR model is also
used for structural inference and policy analysis. In structural analysis, certain
assumptions about the causal structure of the data under investigation
are imposed, and the resulting causal impacts of unexpected shocks or
innovations to specified variables on the variables in the model are summarized.
These causal impacts are usually summarized with impulse response
functions and forecast error variance decompositions.
Eric Zivot: http://faculty.washington.edu/ezivot/econ584/notes/varModels.pdf
Part 10: Time Series Applications [ 56/64]
VAR
y1 (t )  11 y1 (t  1)  12 y2 (t  1)  13 y3 (t  1)  1 x(t )  1 (t )
y2 (t )   21 y1 (t  1)   22 y2 (t  1)   23 y3 (t  1)  2 x(t )   2 (t )
y3 (t )   31 y1 (t  1)   32 y2 (t  1)   33 y3 (t  1)  3 x(t )  3 (t )
(In Zivot's examples,
1. Exchange rates
2. y(t)=stock returns, interest rates, indexes of industrial production,
rate of inflation
Part 10: Time Series Applications [ 57/64]
VAR Formulation
y (t) = y (t-1) + x(t) + (t)
SUR with identical regressors.
Granger Causality: Nonzero off diagonal elements in 
y1 (t )  11 y1 (t  1)  12 y2 (t  1)  13 y3 (t  1)  1 x(t )  1 (t )
y2 (t )   21 y1 (t  1)   22 y2 (t  1)   23 y3 (t  1)   2 x(t )   2 (t )
y3 (t )   31 y1 (t  1)   32 y2 (t  1)   33 y3 (t  1)  3 x(t )   3 (t )
Hypothesis: y2 does not Granger cause y1: 12 =0
Part 10: Time Series Applications [ 58/64]
Impulse Response
y (t) = y (t-1) + x(t) + (t)
By backward substitution or using the lag operator (text, 943)
y (t)  x(t)  x(t-1)   2 x(t-2) +... (ad infinitum)
+ (t)  (t-1)   2 (t-2) + ...
[ P must converge to 0 as P increases. Roots inside unit circle.]
Consider a one time shock (impulse) in the system,  =  2 in period t
Consider the effect of the impulse on y1 ( s ), s=t, t+1,...
Effect in period t is 0.  2 is not in the y1 equation.
 2 affects y2 in period t, which affects y1 in period t+1. Effect is 12  
In period t+2, the effect from 2 periods back is ( 2 )12  
... and so on.
Part 10: Time Series Applications [ 59/64]
Zivot’s Data
Part 10: Time Series Applications [ 60/64]
Impulse Responses
Part 10: Time Series Applications [ 61/64]
GARCH Models: A Model for Time Series
with Latent Heteroscedasticity
Bollerslev/Ghysel, 1974
Part 10: Time Series Applications [ 62/64]
ARCH Model
Part 10: Time Series Applications [ 63/64]
GARCH Model
Part 10: Time Series Applications [ 64/64]
Estimated GARCH Model
---------------------------------------------------------------------GARCH MODEL
Dependent variable
Y
Log likelihood function
-1106.60788
Restricted log likelihood
-1311.09637
Chi squared [
2 d.f.]
408.97699
Significance level
.00000
McFadden Pseudo R-squared
.1559676
Estimation based on N =
1974, K =
4
GARCH Model, P = 1, Q = 1
Wald statistic for GARCH =
3727.503
--------+------------------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------|Regression parameters
Constant|
-.00619
.00873
-.709
.4783
|Unconditional Variance
Alpha(0)|
.01076***
.00312
3.445
.0006
|Lagged Variance Terms
Delta(1)|
.80597***
.03015
26.731
.0000
|Lagged Squared Disturbance Terms
Alpha(1)|
.15313***
.02732
5.605
.0000
|Equilibrium variance, a0/[1-D(1)-A(1)]
EquilVar|
.26316
.59402
.443
.6577
--------+-------------------------------------------------------------