Temple University
Economics 8009
Hypothesis Testing Hwk
1.  The answer to this question is at the end and is in the lecture notes.
2.  First create some indices and get the data.  

Plot the quantities:

2.a.  All of the quanitities are increasing over time.

After the first few years all of the nominal prices are rising, the price of labor the most so.
2.b.  The deflated input prices appear as

Among the deflated prices, only labor can be said to be rising throughout.

2.c. The input output coefficients are 

The input-output coefficients for labor and materials are flat, if not declining. The ratios for capital and energy are quite volatile.

2.  Although there are many ways to verify the accounting relationship between gross output and value added, I have done it using the graph of computed value and the value added series.  This compares favorably with the plot of a 45 degree line.

The slope is not different from 1.
And the intercept is not different from zero.

2.e.  Fit the regression lines.
In the homogeneous model, after normalizing by labor,

For the model in which there are not restrictions on the input coefficients,

2.f. The restriction in 2.e. is equivalent to assuming constant returns to scale in the unrestricted model.

Clearly we should reject the null.
You could do the same test with an F statistic.

3. A return to hedonic prices.

1st Generation
2nd Generation
Whole Sample
Dependent variable

Intercept

ln(MULT)

ln(MEM)

ln(Access)
Coefficients

Residual Sum
of Squares

Now consider a model in which the slopes are equal, but the intercept shifts between the two generations.

3.d.  Testing for stability of slopes and intercepts
i.  Slopes and intercepts are equal between generations: 

Reject the null that all of the coefficients are equal across generations.
ii.  Slopes are equal, but intercepts may be different:
Reject the null of equal intecepts when slopes are constrained to be equal

iii.  Intercepts are unequal, but slopes may be different 

Do not reject the null of equal slopes when the intercepts are allowed to be unequal.
1.  y is a Bernoulli random variable, a single paramter distribution.  
P( y=1 ) = q and P( y =0 ) = 1-q.  To keep things simple suppose there are 5 trials.
The log likelihood is 

The first order condition for a maximum is to take the derivative with respect to q and set the result equal to zero.

This derivative is

 Setting this result equal to zero and solving for q.
q=

The maximum likelihood estimator for q is the sample proportion.
We wish to test the hypothesis

	Ho:   
	H1: qo <>.4
Suppose our sample results are 

and 

=(sum of yt).
So

6. a.  Likelihood Ratio Test
The log-likelihood function evaluated at Ho is

The likelihood evaluated at the sample is 

The LR is asymptotically c2.  In this case it has one degree of freedom.  At the 5% level the critical value is 3.84, so we reject the null.
6.b.  Wald Test
Asymptotically the sample proportion is N(q, q(1-q)/n)

Since W is also c2 with one degree of freedom, the null hypothesis is rejected.
6.c.  Lagrange Multiplier Test
See Greene, pp. 171 

The first and third terms are the derivative of the log likelihood evaluated at the null hypothesis.  The middle term is the inverse of the information matrix (the minus sign is out front), the variance of the first derivative.

The LM statistic is c2 with one degree of freedom, so once again we reject the null hypothesis.