PhD student: Freddy Rojas Cama Homework 1 - ECO 322 Rutgers University

Homework 1Due date: October 15th, 2012; 11.59 PM (ET)

Econometricsby Freddy Rojas Cama

1. A consumption function is an economic relationship between consumption and incomein aggregate terms. This function is estimated by OLS and the results are showed be-low. Speci�cally, the dependent variable is the household consumption (C) and the onlyexplicative variable is the household income (I), those observations are weekly recordedand expressed in real terms. The sample size1 and number of parameters to estimate areN = 15 and K = 2 respectively.

Ct = 28:264 + 0:493945 It + "t

(8:1789) (0:0461)

The standard errors are provided and they are below the parameters or point estimates.

(a) What is the type of data we are dealing with? What is the frequency of those data?.

(b) Is the coe¢ cient related to income or slope (or marginal propensity to consume)signi�cant in statistical terms?.

(c) Construct a 99%, 95% and 90% con�dence intervals for the constant and slope.Interpret those con�dence intervals.

2. Exercise 4.2 in Stock and Watson�s book.

3. Exercise 4.3 in Stock and Watson�s book.

4. Ordinary Least Squares (OLS) delivers the Best Linear Unbiased Estimators if only if(()) certain assumptions are ful�lled. Let�s consider the following linear model

yt = �0 + �1xt + "t

being xt and yt the independent and dependent variables respectively and "t denotes anormal random variable i.i.d, (i.e. identical and independent distribuited) with expecta-tion value or mean equal to zero. Also, xt and "t are independent each other. Considerthe following information in order to have estimates for �0 and �1:

T=50Pt=1

ytxt = 144:70; y = 3:26; x = 0:71;T=50Pt=1

x2t = 74:19

(a) Calculate the BLUE estimates of �0 and �1.

(b) Verify the orthogonality conditions i.ePT=50t=1 b"t = 0 and PT=50

t=1 b"txt = 01Or number of observations.

1

PhD student: Freddy Rojas Cama Homework 1 - ECO 322 Rutgers University

5. A production function is estimated for the US economy using a dataset for year 2011.There are 51 manufacturing �rms in this dataset (N = 51). The estimation is made byOLS

Q = exp(1:37)K0:632L0:452

where b�k = 0:632 and b�l = 0:452: There are 2 independent variables in this model: thelevel of capital and labor skill are denoted as K and L. The dependent variable is thelevel of production (Q). Additionally, we have the following information;

R2 = 0:98 cov(b�k;b�l) = 0:055 std(b�k) = 0:257 std(b�l) = 0:219(a) What is the kind of data we are dealing with? What is the frequency of those data?.

(b) Construct a 99% con�dence interval for �l:

(c) Evaluate if elasticities of labor and capital are identical in statistical terms.2.

(d) Evaluate if there exists evidence of increasing returns to scale (i.e �k + �l > 1) inthis sample.

6. We have the following data y = f25; �1; 4; 0g. Your dear professor wants you toestimate the following model yt = �0 by using the OLS estimator. Speci�cally;

(a) What is the estimate of �0?.

(b) If we add to this model a new variable with same values for each observation, canwe get the OLS estimates? why not?

7. We have the following result (STATA estimation output)

The variables are the state name (state), violent crimes per 100,000 people (crime),percent of population living under poverty line (poverty), and percentage of populationthat are single parents (single). It has 51 observations and this dataset is recorded in aspeci�c period of time. The above ouput shows the results of OLS estimation for a modelwhere the dependent variable is the crime rate and the dependent variables are povertyrate and single status. Answer the following with a true or false statement and supportyour choice.

2Remember thatvar(aX + bY ) = a2var(X) + b2var(Y ) + 2abcov(X;Y )

2

PhD student: Freddy Rojas Cama Homework 1 - ECO 322 Rutgers University

(a) Panel data is the type of data we are dealing with.

(b) The coe¢ cient associated or related to poverty is not signi�cative at 90 and 95%of con�dence level.

(c) The coe¢ cient associated or related to single is not signi�cative at 90 and 95% ofcon�dence level.

8. Show that if we run separated regressions in a model with 2 regressors (one regressionfor each regressor), we can have the same coe¢ cients or point estimates only when theautocorrelation between those 2 regressors is zero.

9. [Playing God and Mortals]. In this game you must choose to be a God or Mortal. If youchoose God, you must generate the following DGP

y = �0 + �1x+ "

where �0 and �1 are the true parameters, x is a random variable distribuited asN��x; �

2x

�and " is distribuited as N

�0; �2

�: You must choose values for the true parameters, �x; �

2x

and �2:[Suggestion: choose those values from the following sets: �0 2 (5; 10), �1 2(�0:45; 0:80) ; �x 2 (3; 10) ; �2x 2 (3; 6) and �2 2 (0:1; 0:45)]. After this parametrization,you must simulate 10000 draws from this DGP. Explain carefully all the steps.

If you choose Mortal, you must look for a God and ask for series y and x: [Warning: Godsdoes not share the DGP with mortals]. Once you have those series, obtain OLS estimatesfor �0, �1 and �

2 (i.e b�0, b�1 and b�2). Report all those estimates, variances and p-values(related to a signi�cant test) for sample sizes 10, 100, 1000 and 10000. Each group mustinclude in its report both results (Mortal and God procedures).

10. We have the following series or data

y x

2 23 34 -23 -3

we consider the following model yi = �0+�1xi+ui where ui is an identical and indepen-dent random variable. Additionally, the mean of ui is equal to zero.

(a) Calculate the OLS estimates (b�0 and b�1)(b) Verify the orthogonality conditions i.e

PNi=1 bui = 0 and PN

i=1 buixi = 0

3