autocorrelation ii
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Lecture 21 1
Econ 140Econ 140
Autocorrelation IILecture 21
Lecture 21 2
Econ 140Econ 140Today’s plan
• Durbin’s h-statistic• Finite Distributed Lags• Koyck Transformations and Adaptive Expectations• Seasonality• Testing in the presence of higher order serially correlated
forms.
Lecture 21 3
Econ 140Econ 140Returning to the Durbin-Watson
• Last time we talked about how to test for autocorrelation using the Durbin-Watson test
• We found autocorrelation in the data in L_20.xls• We used this figure:
2 40
H1
dL dU
H0: =0Reject null Accept null
4-dU 4-dL
Reject nullH1
d = 0.3311.475
Lecture 21 4
Econ 140Econ 140Generalized least squares (3)
• Need an estimate of : we can transform the variables such that:
where:
• Known as Cochrane-Orcutt transformation.• Estimating equation (3) allows us to estimate in the
presence of first-order autocorrelation
(3) * ***ttt ebXaY
1*
ttt YYY
Lecture 21 5
Econ 140Econ 140Problems
1) The model presented by may still have some autocorrelation– the D-W test doesn’t tell us anything about this– we have to retest the model
2) We may lose information when we lag our variables– to get around this information loss, we can use the
Prais-Winsten formula to transform the model:
* ***ttt ebXaY
12*
1
12*
1
1
1
XX
YY
Lecture 21 6
Econ 140Econ 140Problems (2)
3) We might want to include a lagged endogenous variable in the model
– including the lagged endogenous variable Yt-1 biases the Durbin-Watson test towards 2
– this means it’s biased towards the null of no autocorrelation
– in this instance, we’ll use Durbin’s h-statistic (1970):
ttt egYbXaY 11
nvnh
1
v = square of the standard error on the coefficient (g) of the lagged endogenous variable
Lecture 21 7
Econ 140Econ 140Durbin’s h-statistic
• Durbin’s h-statistic is normally distributed and is approximated by the z-statistic (standard normal)
• null hypothesis: H0: = 0– the null can be rejected at the 5% level of significance
• L21.xls has example.• Problems with the h-statistic
– the product nv must be less than one (where n = # of observations)
– if nv 1, the h-statistic is undefined
Lecture 21 8
Econ 140Econ 140A note on consistency
• Model with lagged endogenous variable and first-order serially correlated error may be mis-specified.
Yt = b0 + b1Yt-1 + ut
and ut = ut-1 + et
• If so, presence of first-order serial correlation may induce omitted variable bias.
• Need to include additional lagged endogenous variable term:
Yt = a0 + a1Yt-1 + a2Yt-2 + et
Lecture 21 9
Econ 140Econ 140Why lags?
• This mainly relates to macroeconomic models– economic events such as consumer expenditure,
production, or investment– for instance: consumer expenditure this year may be
related to consumer expenditure last year
• In a general distributed lag model:Yt = a + b0Xt + b1Xt-1 +…+bkXt-k + et
– where k = any large number less than t-2– can eliminate coefficients b1 to bk by using a t-test– number of lags included is ad-hoc
Lecture 21 10
Econ 140Econ 140Problems for OLS
• Lags lead to severe problems for ordinary least squares– loss of information (degrees of freedom)– independent variables (X) are highly correlated [multi-
collinearity problem]
Lecture 21 11
Econ 140Econ 140Why lags are useful
• Psychological reasons: behavior is habit-forming– so things like labor market behavior and patterns of
money holding can be captured using lags
• Technological reasons: a firm’s production pattern
• Institutional: unions
• Multipliers: short run and long run multipliers (how to read finite distributed lags in a model).
Lecture 21 12
Econ 140Econ 140Ad-hoc nature of lags
• What can we do?• Two approaches
– Koyck transformation– Adaptive expectations
• Different implications on the assumptions about economic processes– will end up with the same estimating equation– looking only at the end product, we won’t be able to tell
the Koyck transformation from adaptive expectations
Lecture 21 13
Econ 140Econ 140Koyck transformation
• Model: Yt = a + b0Xt + b1Xt-1 +…+bkXt-k + et
• The Koyck transformation suggests that the further back in time we go, the less important is that factor– for instance, information from 10 years ago vs.
information from last year
• The transformation suggests:j
j bb 0 Where 0 < < 1j = 1,…k
Lecture 21 14
Econ 140Econ 140Koyck transformation (2)
• So,
• Can use the expression for bj to rewrite the model
Yt = a + b0 (Xt + Xt-1 + 2Xt-2 + ….+ kXt-k) + et (4)– this imposes the assumption that earlier information is
relatively less important• Lagging the equation and multiplying it by , we get:
Yt-1 = a + b0 (Xt-1 + 2Xt-2 + ….+ kXt-k) + et-1 (5)
• Subtracting (5) from (4), we getYt = a(1- ) + b0Xt + Yt-1 + vt where vt = et - et-1
and 20201 bbbb
Lecture 21 15
Econ 140Econ 140Koyck transformation (3)
• Why is this transformation useful?– Allows us to take the ad-hoc lag series and condense it
into a lagged endogenous variable– now we only lose one observation due to the lagged
endogenous variable– the given by the estimation gives the coefficient of
autocorrelation• Problem: by construction, we have first-order
autocorrelation– use Durbin h-statistic– but estimating equation might be mis-specified!
Lecture 21 16
Econ 140Econ 140Adaptive expectations
• Another way to approach the problem of the ad-hoc nature of lags
• Can use the example of trying to measure the natural rate of unemployment
• In 1968, Friedman estimated the equation: Yt = a + bXt* + ut
where Xt* = natural rate of unemployment
Lecture 21 17
Econ 140Econ 140Adaptive expectations (2)
• Using adaptive expectations we have thatXt* - Xt-1* = (Xt - Xt-1*)
• Can rewrite the equation: Xt* - (1 - )Xt-1* = Xt
• Using a lag operator where:LXt = Xt-1
L2Xt = Xt-2
Xt* = expectationXt = observed0 < < 1
where
Lecture 21 18
Econ 140Econ 140Adaptive expectations (3)
• We can then rewrite : Xt = (1 - L )Xt*– where = (1- )
• This can be rewritten as:
– now we have the natural rate of unemployment in terms of the observed rate of unemployment
tt XL
X
1
*
Lecture 21 19
Econ 140Econ 140Adaptive expectations (3)
• Substituting into the model we get:
• Upon further multiplication and substitution we arrive at:
– this looks very similar to that for the Koyck transformation
ttt uXL
baY
1
tttt vYXbaY 11
where 11 ttt uuv
Lecture 21 20
Econ 140Econ 140Problems with the approaches
• For the lagged endogenous variables in the ad-hoc lag structure, we are uncertain as to which economic model of agent behavior underlies the estimating equation
• We have 1st-order autocorrelation by the construction of the model– use the Durbin h-statistic
• Yt-1 and et-1 (ut-1) are sure to be correlated [ E(X,e) 0]– this leads to biased estimates – we’ll deal with this using instrumental variables and
simultaneous equations
Lecture 21 21
Econ 140Econ 140Other topics
• Seasonality and the use of dummy variables in time series models.
• Trends and their use in time series models• Testing and correcting in the presence of higher orders of
serial correlation.
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