A More General Model

Equation 9.8 can be expanded and rewritten in the following way:

inf = ві(1 – p) + – в2p Aut-i + p inf— + vt (9.9)

inf = 5 + 5oAut – 5iAut-i + в inft-1 + vt (9.10)

Both equations contain the same variables, but Equation (9.8) contains only 3 parameters while (9.10) has 4. This means that (9.8) is nested within (9.10) and a formal hypothesis test can be

performed to determine whether the implied restriction holds. The restriction is 51 = — в150.1 To

test this hypothesis using gretl you can use a variant of the statistic (6.2) discussed in section 6.1. You’ll need the restricted and unrestricted sum of squared errors from the models. The statistic is

J x F = if H°: 51 = – в15°is true (9Л1)

A = ві(1 – p),50 = 02,Si = – p02,Oi = p

Since J = 1 this statistic has an approximate x2 distribution and it is equivalent to an F test. Note, you will get a slightly different answer than the one listed in your text. However, rest assured that the statistic is asymptotically valid.

For the example, we’ve generated the output:

Chi-square(l): area to the right of 0.112231 = 0.737618 (to the left: 0.262382)

F(1, 85): area to the right of 0.112231 = 0.738443 (to the left: 0.261557)

Because the sample is relatively large the p-values from the F(1,85) and the x2 are very close to one another. Neither is significant at the 5% level.

Подпись: OLS, using observations 1987:3-2009:3 (T Dependent variable: inf = 89) Coefficient Std. Error t-ratio p-value const 0.333633 0.0899028 3.7110 0.0004 d_u -0.688185 0.249870 -2.7542 0.0072 d_u_1 0.319953 0.257504 1.2425 0.2175 inf_1 0.559268 0.0907962 6.1596 0.0000 Mean dependent var 0.783146 S.D. dependent var 0

Подпись:Подпись:

Подпись: Sum squared resid R2 F(3, 85) Log-likelihood Schwarz criterion p Подпись: S.E. of regression Adjusted R2 P-value(F) Akaike criterion Hannan-Quinn Durbin’s h

The estimated model is:

Notice how gretl refers to the parameters-by their variable names. This is possible because the model is linear and there is no ambiguity. Also, Aut_i is referred to as d_u_1. It can get a little confusing, but d_u is the difference and the lag has the usual _1 suffix.

The lagged unemployment rate has a t-ratio of 1.243. It is not significant and it may be worth considering removing it from the model using the omit d_u(-1) statement.

You can also compare nonlinear combinations of parameters from the equations (9.8) and (9.10). To do so you can use gretl to compute the relevant scalars and print them to the screen as shown below in the script:

1 nls inf = beta1*(1-rho) + rho*inf(-1) + beta2*(d_u-rho*d_u(-1))

2 params rho beta1 beta2

3 end nls

4 scalar delta = $coeff(beta1)*(1-$coeff(rho))

5 scalar delta1 = -$coeff(rho)*$coeff(beta2)

6 printf "nThe estimated delta is %.3f and the

7 estimated delta1 is %.3f.n",delta, delta1

In lines 4 and 5 5 and 5i are approximated from the NLS estimated AR(1) regression. the result is

The estimated delta is 0.337 and the estimated delta1 is 0.387.

You can see that these values are actually fairly close to the ones estimated in the unrestricted model, which were 0.334 and 0.320, respectively. Also, /32 is similar to 51 and p is similar to Q. It is no wonder that the hypothesis restrictions are not rejected statistically.

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