Part V Multivariate Time Series Models 429
3.6 Testing linear restrictions on regression coefficients
Consider the linear regression model
yt =α+β1xt1+β2xt2+ut, (3.17)
and assume that it satisfies all the classical assumptions. Suppose now that we are interested in testing the hypothesis
H0:β1+β2=1, against
H1:β1+β2=1.
Let
δ=β1+β2−1, (3.18)
then the test of H0against H1simplifies to the test of H0:δ=0,
60 Introduction to Econometrics
against
H1:δ=0.
The OLS estimator ofδis given by
δˆ = ˆβ1+ ˆβ2−1, and the relevant statistic for testingδ=0 is given by
tδˆ = δˆ−0
Var
δˆ = bˆ1+ ˆb2−1
Var δˆ . where
Var δˆ
=Var βˆ1
+Var βˆ2
+2Cov βˆ1,βˆ2
.
The relevant expressions of the variance-covariance matrix of the regression coefficients are given in relations (2.20)–(2.22).
An alternative procedure for testingδ=0 which does not require knowledge of Covβˆ1,βˆ2
would be to use (3.18) to solve forβ1orβ2in the regression equation (3.17). Solving forβ2, for example, we have
yt =β0+β1xt1+
δ−β1+1
xt2+ut, or
yt−xt2 =β0+β1(xt1−xt2)+δxt2+ut. (3.19) Therefore, the test ofδ=0 againstδ=0 can be carried out by means of a simple t-test on the regression coefficient of xt2in the regression of(yt−xt2)on(xt1−xt2)and xt2.
Example 6 This example describes two different methods of testing the hypothesis of constant returns to scale in the context of the Cobb–Douglas (CD) production function
Yt=AKtαLβt eut, t=1, 2,. . ., T, (3.20) where Yt = Output, Kt = Capital Stock, Lt= Employment. The unknown parameters A,αandβ are fixed, and uts are serially uncorrelated disturbances with zero means and a constant variance.
We also assume that uts are distributed independently of Ktand Lt. The constant returns to scale hypothesis postulates that proportionate changes in inputs (Ktand Lt) result in the same propor- tionate change in output. For example, doubling Ktand Ltshould, under the constant returns to scale hypothesis, lead also to the doubling of Yt. This imposes the following parametric restriction on (3.20):
H0: α+β=1,
which we consider as the null hypothesis and derive an appropriate test of it against the two-sided alternative:
H1: α+β=1.
In order to implement the test of H0against H1, we first take logarithms of both sides of (3.20), which yield the log-linear specification
LYt =a+αLKt+βLLt+ut (3.21)
where
LYt =log(Yt), LKt=log(Kt), LLt =log(Lt)
and a = log(A). It is now possible to obtain estimates ofαandβby running OLS regressions of LYton LKtand LLt(for t=1, 2,. . ., T), including an intercept in the regression. Denote the OLS estimates ofαandβbyαˆ andβ, and define a new parameter,δ, as
δ=α+β−1. (3.22)
The hypothesisα+β=1 againstα+β=1 can now be written equivalently as H0: δ=0,
H1: δ=0.
We now consider two alternative methods of testingδ = 0: a direct method and a regression method. The first method directly focuses on the OLS estimates ofδ, namelyδˆ = ˆα+ ˆβ−1, and examines whether this estimate is significantly different from zero. For this we need an estimate of the variance ofδ. We haveˆ
V(δ)ˆ =V(α)ˆ +V(β)ˆ +2 Cov α,ˆ βˆ
,
where V(ã)and Cov(ã)stand for the variance and the covariance operators, respectively. The OLS estimator of V(δ)ˆ is given by
Vˆ(δ)ˆ = ˆV(α)ˆ + ˆV(β)ˆ +2Cov(ˆ α,β).ˆ The relevant test-statistic for testingδ=0 againstδ=0 is now given by
tδˆ = δˆ
V(ˆ δ)ˆ = αˆ+ ˆβ−1
V(ˆ α)ˆ + ˆV(β)ˆ +2Cov(ˆ α,β)ˆ , (3.23)
and, underδ=0, has a t-distribution with T−3 degrees of freedom. An alternative method for testingδ=0 is the regression method. This starts with (3.21) and replacesβ(orα) in terms ofδ
62 Introduction to Econometrics
andα(orβ). Using (3.22) we have
β=δ−α+1.
Substituting this in (3.21) forβnow yields
LYt−LLt =a+α(LKt−LLt)+δLLt+ut, or
Zt=a+αWt+δLLt+ut, (3.24)
where Zt =log(Yt/Lt)=LYt−LLtand Wt =log(Kt/Lt)= LKt −LLt. A test ofδ =0 can now be carried out by first regressing Zton Wtand LLt(including an intercept term), and then carrying out the usual t-test on the coefficient of LLtin (3.24). The t-ratio ofδin (3.24) will be identical to tδˆdefined by (3.23). We now apply the two methods discussed above to the historical data on Y, K, and L used originally by Cobb and Douglas (1928), covering the period 1899–1922.
The following estimates ofα,ˆ βˆ and of the variance covariance matrix of(α,ˆ β)ˆ can be obtained:
ˆ
α=0.23305, βˆ =0.80728,
⎡
⎣ Vˆ(α)ˆ Cov α,ˆ βˆ Cov
α,ˆ βˆ
Vˆ(β)ˆ
⎤
⎦=
0.004036 −0.0083831
−0.0083831 0.021047
.
Using the above results in (3.23) yields
tδˆ= 0.23305+0.80728−1
√0.004036+0.021047−2(0.0083831) =0.442. (3.25) Comparing tδˆ =0.442 and the 5 per cent critical value of the t-distribution with T−3=24− 3 =21 degrees of freedom (which is equal to 2.080), it is clear that since tδˆ =0.442 <2.080, then the hypothesisδ=0 orα+β =1 cannot be rejected at the 5 per cent level. Implementing the regression approach, we estimate (3.24) by OLS and obtain estimates for the coefficients of Wtand LLtof 0.2330(0.06353) and 0.0403(0.0912), respectively. (The figures in brackets are standard errors.) Note that the t-ratio of the coefficient of the LL variable in this regression is equal to 0.0403/0.0912=0.442, which is identical to tˆδas computed in (3.25). It is worth noting that the estimates ofαandβ, which have played a historically important role in the literature, are very
‘fragile’, in the sense that they are highly sensitive to the sample period chosen in estimating them.
For example, estimating the model (given in (3.21)) over the period 1899–1920 (dropping the observations for the last two years) yieldsαˆ =0.0807(0.1099)andβˆ =1.0935(0.2241).