Problem Set B (Fall 2016)
In this problem set, we cover multiple regression, F-tests, t-tests, functional forms, and more.
The Cobb-Douglas Celebration.
1. In production theory, there is the celebrated Cobb-Douglas Production Function. It can
be expressed as:
Y X X e
ui
i i i
? ? ? 2 3
1 2 3
? , where Y=output, X 2 =labor input, and X3 =capital input.
From this equation, it is clear that the relationship between output and the two inputs is
nonlinear.
a) Take the natural logs of both sides of the equation
.what do you get?
______________________________________________________________________
If we let ? ? 1 0 ln ? , the model should now be linear in the parameters, ? 0
, ? 2
, and ? 3
, and is therefore a linear regression model.
Gujarati describes to us that ? 2
is the partial elasticity of output with respect to the labor
input, that is, it measures the percentage change in output for a 1% change in the labor input,
holding the capital input constant.
b) How is ? 3
generally interpreted as?
An important feature of the Cobb-Douglas production function is the information it gives about
returns to scale-that is, the response of output to a proportionate change in inputs. This
would be important information for a production process. Consider, for instance, the
manufacturing of iPhones, or the Surface Pro computer.
The sum ? 2
+ ? 3
gives us this information. If this sum is 1, then there are constant returns to
scale, that is doubling the inputs will double the output. If the sum is < 1, then there are
decreasing returns to scale, that is, doubling the inputs will less than double the output. Finally,
if the sum is > 1, there are increasing returns to scale, that is, doubling the inputs will more than
double the output.
Take the following data from the agricultural sector of Taiwan for 1958 1972 and run the
Cobb-Douglas production function using your specification above in part a).
Year Real gross product Labor days Real capital input
1958 16607.7 275.5 17803.7
1959 17511.3 274.4 18096.8
1960 20471.2 269.7 18271.8
1961 20932.9 362.0 19167.3
1962 20406.0 267.8 19647.6
1963 20831.6 275.0 20803.5
1964 24806.3 283.0 22076.6
1965 26465.8 450.7 23445.2
1966 27403.0 307.5 24939.0
1967 28628.7 303.7 32713.7
1968 29904.5 304.7 29957.8
1969 27508.2 298.6 31585.9
1970 29035.5 295.5 33474.5
1971 29281.5 299.0 59821.8
1972 24535.8 288.1 41794.3
c) Write the estimated equation below:
______________________________________________________________
d) Interpret the coefficients, b2
and b3
in words!
_____________________________________________________________________
_____________________________________________________________________
e) What is the nature of the returns to scale?
===============================================================
An F-test: the collective, a multiple regression
2. Consider the data in the excel file called caschool and the accompanying data description key called californiatestscores.
a. run a the regression of testscores on the independent variables student-teacher ratios and also PctEL. Report your results below using the SAS output (copy and paste).
b. perform the joint hypothesis to test whether the independent variables are having an influence collectively on the dependent variables. you only need to write out the null and alternative hypothesis, produce the appropriate test statistic (should be on the SAS output), and make a decision using the p-value.
==============================================================
Purify: The residuals are the gateway to understand partial effects.
3. Look at the data below. For the entire model specification of
kwhi = b1 + b2(pelec)2i + b3(gnp)3i + ei,
purify the model of gnps influence (b3) and present your results as we did in class. I want to see SAS output that 1) points out the value of the coefficient on pelec from the full model (as we did in class) and 2) points out the value for that same coefficient from the purified model. Also, explain what each set of residuals mean (in words) as we did in class. For instance, the residuals generally represent everything that potentially influences the dependent variable besides the independent variable.
kwh
pelec
gnp
330
3.12
579.4
356
3.09
600.8
396
3.01
623.6
424
2.97
616.1
497
2.75
657.5
546
2.61
671.6
576
2.57
683.8
588
2.59
680.9
647
2.5
721.7
689
2.65
737.2
723
2.63
756.6
778
2.55
800.3
833
2.47
832.5
896
2.38
876.4
954
2.29
929.3
1035
2.16
984.8
1099
2.09
1011.4
1203
1.97
1058.1
1314
1.88
1087.6
1392
1.83
1085.6
1470
1.84
1122.4
1595
1.86
1185.9
1712.91
1.85
1254.3
1705.92
2.16
1246.3
1747.09
2.32
1231.6
1855.25
2.33
1298.2
1948.36
2.44
1369.7
2017.92
2.45
1438.6
2071.1
2.44
1479.4
2094.45
2.65
1474
2147.1
2.79
1502.6
2086.44
2.96
1479.98
2150.96
2.92
1534.68
2278.37
2.92
1639.32
==============================================================
More Testing with multiple regressions
4. Take the data below and run a regression model of Y on X2 and X3 in SAS.
Personal consumption expenditure and personal disposable income in the United States, 1956-
1970, billions of 1958 dollars.
PCE, Y PDI, X2 Time, X3
220.6 309.3 1956=1
288.1 316.1 1957=2
290.0 313.5 1958=3
307.3 333.0 1959=4
316.1 340.3 1960=5
322.5 350.5 1961=6
328.5 367.2 1962=7
353.3 381.2 1963=8
373.7 416.0 1964=9
397.7 434.8 1965=10
418.1 458.9 1966=11
430.1 477.5 1967=12
433.5 499.0 1968=13
469.1 513.5 1969=14
476.9 536.7 1970=15
a) Report your results below:
Yi
^
= __________ + _________ X2i
+ _________ X 3i
b) We can look at the individual t-tests to conduct hypothesis tests about the individual
partial regression coefficients. Specifically, we will focus on the size of the tcomputed given to us in
the output for each of the slope coefficients b2 and b3. Also, we will look at the reported pvalues
for each of the coefficients. In a nutshell, when we do hypothesis testing we can define a
rare event arbitrarily by setting a threshold level for our p-value. If our p-value (whats reported
to you in SAS) falls below that point well reject the null hypothesis. We call such results
statistically significant. The threshold is called an alpha level.
Let the null and alternative hypothesis for B2 be
.
H0: B2 = 0 (says that holding X3 constant, personal disposable income has no linear influence on
personal consumption expenditure,Y)
H1: B2 ? 0 (there is an influence)
Now, you do the null and alternative hypothesis for B3:
_______________________________________________________________________
_______________________________________________________________________
Using the p-value method, are the reported coefficients for b2 and b3 statistically significant at the .05 (a.k.a. 5% alpha level) alpha level?
==============================================================
Functional forms.
5. Pendulum. A student experimenting with a pendulum counted the number of full swings the pendulum made in 20 seconds for various lengths of string. Her data are shown below.
Length (in.)
6.5
9
11.5
14.5
18
21
24
27
30
37.5
Number of Swings
22
20
17
16
14
13
13
12
11
10
a) Is a linear model appropriate for this data when trying to predict the number of swings using length? Explain how you came to your answer (hint: using SAS, can you look at the scatterplot with swings on the vertical and length on the horizontal?). Finally, produce a scatterplot that plots the residuals from the linear model against the independent variable length, comment on what you see.
b) Using SAS, re-express (using the suggestions below) the data to straighten the scatterplot. Run residual plots for each and comment on what you see. Label your SAS output graphs according the attempts below.
Try the natural log of swings.
Try the square root of swings.
Try the reciprocal of swings.
Try 1 over the square root of swings.
c) Based on what youve observed from the residual plots, choose an appropriate model, and then predict the number of swings for a pendulum with a 4 string.
==============================================================
More on Functional forms.
6. Consider the data set called baseball_Hilmer_PSB.
a. Plot salary (y) against experience (x). Describe what you see.
b. Estimate the functional form (run the regression) that is:
salaryi = b1 + b2(experience)2i + ei
Plot the residuals against experience from this regression using the output statement in SAS. Comment.
c. Estimate the functional form (run the regression) that is:
salaryi = b1 + b2(experience)2i + b3(experience2)3i + ei
Plot the residuals against experience from this regression using the output statement in SAS. Comment.
d. Predict salary for a player with 14.33 years of experience.
e. Finally, for fun, estimate the semi-log functional form that is:
ln(salary)i = b1 + b2(experience)2i + b3(homeruns)3i + ei
==============================================================
Nonlinear in the variables.
7. As an extension of functional forms, lets estimate a polynomial model consider the cost-output data below. We actually did this as a class exercise, but I want you to examine and comment on the behavior of the sample residuals in the steps below.
Y
X
193
1
226
2
240
3
244
4
257
5
260
6
274
7
297
8
350
9
420
10
a. plot y against x and explain what you see.
b. run a standard OLS and obtain the residuals. plot the residuals against x and explain what you see. SAS will generate a default residual plot (see below for an example) but remember in lecture that we can also do us proc gplot after we output the residuals to a temporary data set.
c. run a cubic cost equation, that is, estimate the following using proc reg:
y-hat = b1 + b2Xi + b3Xi2 + b4Xi3 + ei
obtain the residuals and plot again as in step 2. explain what you see.
Multicollinearity.
8. This is a question that we will do partly in lecture together and partly on your own. But you will be given the directions for this question in lecture. It is a regression of SAT scores on a number of independent variables.
Roses and the demand for them
9. This is the Roses exercise that we did in lecture. Put your output here
For a custom paper on the above topic or any other topic, place your order now!
What Awaits you:
On-time delivery guarantee
Masters and PhD-level writers
Automatic plagiarism check
100% Privacy and Confidentiality
High Quality custom-written papers
<
