Part V Multivariate Time Series Models 429
5.6 Cochrane–Orcutt iterative method
This estimation method employs the Cochrane and Orcutt (1949) iterative procedure to com- pute ML estimators of β under the assumption that the disturbances, ut, follow the AR(p) process
ut = p
i=1
φiut−i+ t, t∼N(0,σ2), t=1, 2,. . ., T, (5.43)
with ‘fixed initial’ values. The fixed initial value assumption is the same as treating the values, y1, y2,. . ., ypas given or non-stochastic. This procedure in effect ignores the possible contribu- tion of the distribution of the initial values to the overall log-likelihood function of the model.
Once again the primary justification of treating initial values as fixed is asymptotic and is plau- sible only when (5.43) is stationary and T is reasonably large (see Pesaran and Slater (1980, Section 3.2), and Judge et al. (1985) for further discussion).
The log-likelihood function for this case is defined by
LLCO(θ)= −(T−p)
2 log(2πσ2)− 1 2σ2
T t=p+1
2
t +c, (5.44)
whereθ =(β,σ2,φ)withφ=(φ1,φ2,. . .,φp). Notice that the constant term c in (5.44) is undefined, and is usually set equal to zero. The Cochrane–Orcutt (C-O) method maximizes LLCO(θ), or equivalently minimizest=p+1T 2t with respect toθby the iterative method of ‘suc- cessive substitution’. Each iteration involves two steps: in the first step, LLCOis maximized with respect toβ, takingφas given. In the second step,βis taken as given and the log-likelihood func- tion is maximized with respect toφ. In each of these steps, the optimization problem is solved by running OLS regressions. To start the iterationsφis initially set equal to zero. The iterations are terminated if
p i=1
$$$φ˜i,(j)− ˜φi,(j−1)$$$<p/1000, (5.45)
whereφ˜i,jandφ˜i,(j−1)stand for estimators ofφiin the jthand(j−1)thiterations, respectively.
The estimator ofσ2is computed as ˆ σ2=
T t=p+1
˜2t/(T−p−k), (5.46)
where˜t, the adjusted residuals, are given by
˜t = ˜ut− p
i=1
φ˜iu˜t−i, t=p+1, p+2,. . ., T, (5.47)
where
˜
ut =yt− k
i=1
β˜ixit, t=1, 2,. . ., T. (5.48)
As before, the symbol∼on top of an unknown parameter stands for ML estimators (now under fixed initial values). The estimator of σ2 in (5.46) differs from the ML estimator, given by
˜
σ2=T
t=p+1˜2t/(T−p). The estimatorσˆ2allows for the loss of degrees of freedom associ- ated with the estimation of the unknown coefficients,β, and the parameters of the AR process, φ. Notice also that the estimator ofσ2is based on T−p adjusted residuals, since the initial values y1, y2,. . ., ypare treated as fixed.
The adjusted fitted values,y˜t, in the case of this option are computed as
˜
yt = ˆE(yt$$yt−1, yt−2,. . .; xt, xt−1,. . .) =yt− ˜t, (5.49) for t=p+1, p+2,. . ., T. Notice that the initial valuesy˜1,y˜2,. . .,y˜p, are not defined.
In the case where p = 1, Microfit also provides a plot of the concentrated log-likelihood function in terms ofφ1, defined by
LLCO(φ˜1)= −(T−1)
2 [1+log(2πσ˜2)], (5.50)
where
˜ σ2=
T t=2
˜2t/(T−1),
and˜t = ˜ut− ˜φ1u˜t−1.
5.6.1 Covariance matrix of the C-O estimators
The estimator of the asymptotic variance matrix ofφ˜ =(β˜,φ˜)is computed as V(ˆ φ)˜ = ˆσ2
' X˜∗X˜∗ X˜∗S SX˜∗ SS
(−1
, (5.51)
whereX˜∗is the(T−p)×k matrix of transformed regressors2
2A typical element ofX˜∗is given by
˜ x∗jt=xjt−
p i=1
˜
ρixj,t−i t=p+1, p+2,. . ., T, j=1, 2,. . ., k.
108 Introduction to Econometrics
X˜∗= p
i=1
φ˜iX−i, (5.52)
and S is an(T−p)×p matrix containing the p lagged values of the C-O residuals,u˜t, namely
S=
⎛
⎜⎜
⎜⎝
˜
up u˜p−1 . . . u˜1
˜
up+1 u˜p . . . u˜2
... ... ... ...
˜
uT−1 u˜T−2 . . . u˜T−p
⎞
⎟⎟
⎟⎠. (5.53)
The unadjusted residuals,u˜t, are already defined by (5.48). The above estimator of the variance matrix ofβ˜andφ˜is asymptotically valid even if the regression model contains lagged dependent variables.
Example 12 Consider the regression equation
st =α0+α1st−1+α2log yt+α3
t−et
+ut, (5.54)
where stis the saving rate,log ytis the rate of change of real disposable income,tis the rate of inflation, andetare the adaptive expectations oft, and
ut =φ1ut−1+ t. (5.55)
Equation (5.54) is a modified version of the saving function estimated by Deaton (1977).3In the following, we use an approximation ofetby a geometrically declining distributed lag function of the UK inflation rate (see Lesson 10.12 in Pesaran and Pesaran (2009) for details). Figure 5.1 shows the log-likelihood profile for different values ofφ1, in the range[−0.99, 0.99]. The log-likelihood function is bimodal at positive and negative values ofφ1. The global maximum of the log-likelihood is achieved forφ1 <0. Bimodal log-likelihood functions frequently arise in estimation of models with lagged dependent variables subject to a serially correlated error process, particularly in cases where the regressors show a relatively low degree of variability. The bimodal problem is sure to arise if apart from the lagged values of the dependent there are no other regressors in the regression equation. Table 5.1 reports maximum likelihood estimation of the model in the Cochrane–Orcutt method. The iterative algorithm has converged to the correct estimate ofφ1(i.e.φˆ1= −0.22838) and refers to the global maximum of the log-likelihood function given by LL(φˆ1 = −0.22838)
=445.3720. Notice also that the estimation results are reasonably robust to the choice of the initial estimates chosen forφ1, so long as negative or small positive values are chosen. However, if the iter- ations are started fromφ(0)1 =0.5 or higher, the results in Table 5.2 will be obtained. The iterative process has now converged toφˆ1=0.81487 with the maximized value for the log-likelihood func- tion given by LL(φˆ1=0.81487)=444.3055, which is a local maximum. (Recall from Table 5.1
3Note, however, that the saving function estimated by Deaton (1977) assumes that the inflation expectationsetare time invariant.
450
440
430
420
410
–0.99 –0.5 0 0.5
Parameter of the autoregressive error process of order 1 Plot of the Concentrated Log-likelihood Function
0.99
Figure 5.1 Log-likelihood profile for different values ofφ1.
that LL(φˆ1 = −0.22838)=445.3720.) This example clearly shows the importance of exper- imenting with different initial values when estimating regression models (particularly when they contain lagged dependent variables) with serially correlated errors. (For further details see Lesson 11.6 in Pesaran and Pesaran (2009).)
Table 5.1 Cochrane–Orcutt estimates of a UK saving function
Cochrane–Orcutt methodAR(1) converged after 3 iterations Dependent variable is S
140 observations used for estimation from 1960Q1 to 1994Q4
Regressor Coefficient Standard Error T-Ratio[Prob]
INPT −.0032323 .0041204 −.78448 [.434]
S(−1) .99250 .040347 24.5989 [.000]
DLY .66156 .060082 11.0111 [.000]
DPIE .31032 .093382 3.3231 [.001]
R-Squared .76673 R-Bar-Squared .75977
S.E. of Regression .010004 F-Stat. F(4,134) 110.1102 [.000]
Mean of Dependent Variable .096441 S.D. of Dependent Variable .020696
Residual Sum of Squares .013412 Equation Log-likelihood 445.3720
Akaike Info. Criterion 440.3720 Schwarz Bayesian Criterion 433.0179
DW-statistic 1.9615
Parameters of the autoregressive error specification U= −.22838*U(-1)+E
( −2.5135) [.013]
t-ratio(s) based on asymptotic standard errors in brackets
110 Introduction to Econometrics
Table 5.2 An example in which the Cochrane–Orcutt method has converged to a local maximum Cochrane–Orcutt methodAR(1) converged after 7 iterations
Dependent variable is S
140 observations used for estimation from 1960Q1 to 1994Q4
Regressor Coefficient Standard Error T-Ratio[Prob]
INPT .075353 .0098576 7.6441 [.000]
S(−1) .19990 .084385 2.3689 [.019]
DLY .55758 .052907 10.5388 [.000]
DPIE .45522 .10271 4.4322 [.000]
R-Squared .76312 R-Bar-Squared .75605
S.E. of Regression .010081 F-Stat. F (4,134) 107.9234 [.000]
Mean of Dependent Variable .096441 S.D. of Dependent Variable .020696
Residual Sum of Squares .013619 Equation Log-likelihood 444.3055
Akaike Info. Criterion 439.3055 Schwarz Bayesian Criterion 431.9514
DW-statistic 2.2421
Parameters of the autoregressive error specification U= .81487*U(−1)+E
( 16.1214) [.000]
t-ratio(s) based on asymptotic standard errors in brackets