Q1

i

The model that the researcher tests can be described as

ln(wage)t=k*ln(wage)(t-1)+ut

Where ut is a random error variable, normally distributed with mean 0

If this is a random walk, k=1.

However, to test that, it is more convenient to express it in difference form, and test

whether the coeficient(delta) is 0. For empirical reasons, constant is added

ln(wage)t-ln(wage)(t-1)=a+delta*ln(wage)(t-1)+ut

Ho:delta=0

H1:delta<>0

Yt-Y(t-1)=a+delta Y(t-1) + ut

if the absolute value of the computed DF statistics exceeds critical value, reject H0

                      Unit root tests for variable LNREAL

      The Dickey-Fuller regressions include an intercept but not a trend

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 130 observations used in the estimation of all ADF regressions.

 Sample period from 1960Q4 to 1993Q1

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        Test Statistic      LL           AIC           SBC           HQC

 DF         -.42894      360.3673      358.3673      355.4998      357.2022

 95% critical value for the augmented Dickey-Fuller statistic =  -2.8837

|DF|<|Critical value|

Do not reject H0, it can be a random walk.

ii

Ho:delta=0

H1:delta<>0

Yt-Y(t-1)=a+delta Y(t-1) + ut

if the absolute value of the computed DF statistics exceeds critical value, reject H0

                        Unit root tests for variable LO

      The Dickey-Fuller regressions include an intercept but not a trend

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 130 observations used in the estimation of all ADF regressions.

 Sample period from 1960Q4 to 1993Q1

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        Test Statistic      LL           AIC           SBC           HQC

 DF         -1.1729      413.7883      411.7883      408.9208      410.6231

 95% critical value for the augmented Dickey-Fuller statistic =  -2.8837

|DF|<Critical value

Do not reject H0, it can be a random walk.

iii

Ho:delta=0

H1:delta<>0

Yt-Y(t-1)=a+delta Y(t-1) + ut

if the absolute value of the computed DF statistics exceeds critical value, reject H0

                        Unit root tests for variable LC

      The Dickey-Fuller regressions include an intercept but not a trend

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 130 observations used in the estimation of all ADF regressions.

 Sample period from 1960Q4 to 1993Q1

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        Test Statistic      LL           AIC           SBC           HQC

 DF         -2.9646      352.2209      350.2209      347.3534      349.0557

 95% critical value for the augmented Dickey-Fuller statistic =  -2.8837

|DF|>Critical value

Reject H0, we dont have a random walk

iv

Although two variables seem to follow the random walk, a linear combination of them does not do so. Thus although we cannot predict which way the real wage and output per head go we can predict the movements in cost. This is probably because real wage and output depend so heavily on technological improvements, that are arguably ranom, however, costs depend other things as well, on trade cycle, etc. Thus one can predict how much the cost is going to be on the next period, despite the fact that the magnitudes of the components that make up costs, wages and output, cannot be determined.

Q2

X1 is wages, X2 prices, X3 capital/employment ratio and X4 unemployment rate.

                       Ordinary Least Squares Estimation

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 Dependent variable is X1

 32 observations used for estimation from 1959 to 1990

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 Regressor              Coefficient       Standard Error         T-Ratio[Prob]

 C                         .035507             .14685             .24178[.811]

 X1(-1)                     .44479             .13727             3.2402[.003]

 X2                         .95014            .064631            14.7011[.000]

 X2(-1)                    -.40682             .14319            -2.8411[.009]

 X3                         .52885             .13224             3.9991[.000]

 X4                       -.040442            .013155            -3.0744[.005]

 X4(-1)                  -.0057808            .013578            -.42576[.674]

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 R-Squared                     .99988   R-Bar-Squared                   .99985

 S.E. of Regression           .012351   F-stat.    F(  6,  25)   35372.8[.000]

 Mean of Dependent Variable    .90632   S.D. of Dependent Variable      1.0220

 Residual Sum of Squares     .0038134   Equation Log-likelihood        99.1536

 Akaike Info. Criterion       92.1536   Schwarz Bayesian Criterion     87.0235

 DW-statistic                  1.9135   Durbin's h-statistic      .38824[.698]

* A:Serial Correlation*CHSQ(   1)=   .10516[.746]*F(   1,  24)=  .079127[.781]*

* B:Functional Form   *CHSQ(   1)=   1.4509[.228]*F(   1,  24)=   1.1399[.296]*

* C:Normality         *CHSQ(   2)=   .16545[.921]*       Not applicable       *

* D:Heteroscedasticity*CHSQ(   1)=   .84065[.359]*F(   1,  30)=   .80937[.375]*

All the regressors, except lagged unemployment and the constant term, are significant. The R squared is very high. Which means the regression model is very good. We had no previous reason to expect the wages to be determined by a constant anyway. Also there is no problem with heteroscedacity, autocorrelation, different functional form or nonnormality of errors

ii

It is all given in the previous table.

So for example, 1% change in X1 will lead to 0.44% change in level of wages

Roughly, to test for the unity a value t=(b-1)/(se)<2. B is the coefficient we are testing and se its standard error

It is clear that only the prices can possibly be unity, test statistics is

=(0.95014-1)/0.064631

=

-0.771456422

meaning that x2 can be unity

iii

Original RSS

0.003813

1. Restricted by leaving lagged UE out

 Residual Sum of Squares     .0038410   Equation Log-likelihood        99.0380

2. Restricted by including only lagged UE out

 Residual Sum of Squares     .0052551   Equation Log-likelihood        94.0227

3. Restricted by including the difference

 Residual Sum of Squares     .0051453   Equation Log-likelihood        94.3604

I will use the restricted least squares F test approach for which F=RSS(r)-RSS(ur)(n-k)/RSSur*m

Follows the F distribution with m, n-k df. Ur means unrestricted. M is number of linear restrictions, 1

k number of parameters in ur=7, n=32, df=25, F critical at 5% level =4.25

For restriction

1

F=

0.180940

For restriction

2

F=

9.451539

For restriction

3

F=

8.731709

So I reject restrictions 2 and 3, restriction one, leaving lagged UE out is reasonable

iv

Chow test

H0:structural stability

H1:break in the structure

* F:Chow Test         *CHSQ(   6)=   2.4940[.869]*F(   6,  20)=   .41567[.860]*

Do not reject H0, structuraly stable.