Solution to the take home exam for ECON 3150/4150

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1 Solution to the tae home exam for ECO 350/450 Jia Zhiyan and Jo Thori Lind April 2004 General comments Most of the copies we ot were quite ood, and it seems most of you have done a real effort on the problem set. We will first ive some eneral comments on common mistaes and misunderstandins, and then present solutions to problems 2 and 3. Particularly problem 3 is very much a discussion question, so there is no such thin as a correct answer. But the suested solution attempts to include most required elements. There are also other ways of taclin the problem that are equally satisfactory. We have not included a solution to problem as it is very much explained in the boo. Some of you don t discuss your results. You should always try to add a line or two of comments after a test, and when you et a reression try to discuss the main results a bit. There is some confusion about the requirements for OLS to be unbiased. When the reressors are non-stochastic, all that is needed is that the true model is linear, and that the residual has expectation zero. Assumptions on the variance and covariance are only necessary to derive variances of the estimators and for BLUE to hold. Some of you hardly define what BLUE is. Remember to answer the question! In problem 2, it is sometimes hard to follow what you do. Try to add some words to motivate what you re tryin to do. In problem 3, when ased to set up an econometric model, this includes statin assumptions on the residual.

2 Some of you are tempted to use the central limit theorem as we have a lot of observations. This is nice. However, the CLT does not say that each residual becomes normal when is lare. However, even if each residual is non-normal but with identical distribution, the estimators for the parameters, which in a sense are averaes of the residuals, become normally distributed. When testin whether two reression coefficients are different, there are three approaches. One is to find the covariance between the estimates (B not between the variables) and construct a standard t-test. The second is to rephrase the reression as we do in the solution of problem 3 below. The last is to use an F-test and comparin RRS R and RSS U.

3 Problem 2 (answers in bold) Suppose we are interested in estimatin the followin reression model: () Y = α0 + αx +ε 2 σ if i=j and = var( ε, ε j ) = ε 0 otherwise and E( ε ) = 0 for all i and Unfortunately, the variables Yij and on roup averaes, which are defined by * i= has identical size.) So instead, we can only estimate the followin model: X ij are not directly observable. We have only access to data * Y = Y / and X = X / (Each roup i= (2) Y = β + β X + e * * 0 Denote the OLS estimates of β obtained from reression (2) by ˆβ Denote the OLS estimator of α obtained from reression () by ˆ α. Question. Clarify the connections between the reressions () and (2). Is ˆβ an unbiased estimator of α? Answer: Sum up () for each roup, we have Y = α + α X + ε 0 i= i= i= Divide by at both sides Denote e = i= It is easy to verify that Y / = α + α X / + ε / 0 i= i= i= ε /, and use definition of i= * Y and Ee ( ) = E( ε )/ = 0 cov( e, e) = cov( ε /, εi / ) = i= i= * X, we have (2) 2 σ ε / if = 0 otherwise

4 So (2) also satisfy the assumption for OLS. Usin the OLS formula, we have * * * * * ( X X) Y ( X X)( α0 + αx + e) ( X X) e ˆ β = = = α * 2 * 2 + * 2 ( X X) ( X X) ( X X) so we have * ( X X) E( e) * 2 ( X X) E( ˆ β ) = α + = α * ( X X) e 2 var( e) σ = = = ε * 2 * 2 * 2 ( X X) ( X X) ( X X) var( ˆ β ) var( ) Question 2. Calculate the variances of ˆ α and ˆβ. Which estimator of α do you prefer. Explain the reason for your choice Similar to above: we have ˆ α ( X X) ε i == α+ 2 ( X X) i One thin to note here is the double summation X. The outer ( subscript ) i= summation is over different roups, the inner summation (subscript i) is over the observations within the same roup. You can thin it as summin over the followin table row by row: First individual of the roup (i=) Group (=) X Group 2 (=2) X2 Last individual of the roup (i=) X X X X X 2 X Last roup (=G) XG X X G X Easy to show that ˆ α is unbiased estimator of α. And the variance of ˆ α is

5 2 σ ε ˆ 2 ( X X ) i var( α ) == Since both ˆ α and ˆβ is unbiased, we will choose the one with smaller variance. Compare the formulae for variance of ˆ α and ˆβ, we see that ˆ α ˆ β iff X X X X Usin the hints iven, we see that 2 * 2 var( ) var( ) ( ) ( ) i ( X X) = ( X X + X X) 2 * * 2 i i = (( X X ) + ( X X)) i i * * 2 = + + * 2 * 2 * * (( X X ) ( X X) 2( X X )( X X)) = ( X X ) + ( X X) + (( X X) ( X X * 2 * 2 * * i i = ( X X ) + ( X X) * 2 * 2 i * ( X X ) =0 for all. i ote that in the last step we use that fact that So which proves that var( ˆ α ˆ ) var( β). X X X X = X X 0 2 * 2 * 2 ( ) ) ( ) ( ) i i )) Question 3. Will you chane your conclusion if you are informed that the reressor X is constant within each roup but varies between roups i.e. X =X j. Answer: if X =X j then X = X * for all, so ( X X) ) ( X X) 2 * i 2 =0 which means that var( ˆ α ˆ ) = var( β). And there is no loss of efficiency for usin roup averae instead of individual data.

6 Problem 3 a) Consider the model w i = α + β lo y i + ε i, i =,..., where w is the expenditure share on rice, y total consumer expenditure, and ε i a random error term. I will assume that. ε i (0, σ 2 ) 2. cov (ε i, ε j ) = 0, i j. 3. y i is non-stochastic This will ive me an econometric model where OLS is unbiased and BLUE, and where the usual formulae ive correct expressions for the variances. otice that assumption ) implies that ε i both has expectation zero and a constant variance. Usin the supplied data, I estimated this model. The results are as follows: Constant Ltotexp sma RSS R^ F(,2245) = [0.000]** lo-lielihood DW.5 no. of observations 2247 no. of parameters 2 mean(w_rice) var(w_rice) This shows that there is a neative relationship between lo of total expenditure and the budet share on rice. A one percentae increase in income ives a increase in the budet share on rice. The t-value of -7.5 shows that this is a stronly snificant result that and we would reject the null hypothesis of no relationship between total expenditure and the budet share on rice at all usual levels of snificance. The R 2 of 0.2 is relatively low, so there

7 is a reat deal of variation that is not explained by expenditure. As this is household micro data, it is not unusual to et relative low values of R 2 thouh. b) An Enel curve ives the relationship between total expenditure and expenditure on a iven ood. If we let x denote expenditure on rice, the Worin-Leser specification ives rise to an Enle curve of the form x = αy + βy lo y Pluin in the estimates from the estimation, we et a curve as shown on the next pae. Each data point is also shown to ive an impression of the fit of the model. This ives an increasin but concave curve. At very hh incomes it starts to decrease, but almost no households are on this part of the curve. This shows that rice is a normal non-luxury ood (for the very rich, it is an inferior ood). c) We first construct a variable hhsize that ives the number of person livin in the household. Let us assume that the same assumptions that under a) applies. We include this new variable in the reression, which yield the followin output: Constant Ltotexp hhsize sma RSS R^ F(2,2244) = [0.000]** lo-lielihood DW.05 no. of observations 2247 no. of parameters 3 mean(w_rice) var(w_rice) We notice that household size has a positive and snificant impact on the budet share for food. An additional person in the household is estimated to imply a increase in the budet share on rice. This is to some extent evidence that lare households are more costly

8 Enel curve showin the relationship between total consumer expenditure and expenditure on rice.

9 to run than smaller households. Also, the coefficient on expenditure is now larer. This is probably due to multicollinearity as it is natural to assume that larer households also have larer expenditures. Finally, the fit of the model as measured by the R 2 increases substantially. It seems plausible that children and adults have different needs and hence different impacts on rice consumption. To see whether this is the case, we separate hhsize into adults and children. The new estimates are as follows: Constant Ltotexp adults children sma RSS R^ F(3,2243) = 6 [0.000]** lo-lielihood DW.05 no. of observations 2247 no. of parameters 4 mean(w_rice) var(w_rice) We notice that the budet share on rice is affected twice as much but an additional child as by an additional adult. To test whether this difference is snificant, notice that we can write the relationship as w = α + β lo y + γ children + γ 2 adults The null hypothesis is γ =γ 2, which we test versus the alternative of γ γ 2. Under the null, we can write the model as w = α + β lo y + γ (children+adults) oticin that children+adults=hhsize, a convenient way to test the hypothesis is to include hhsize and children (or adults) as explanatory variables. Under the null, the coefficient on children should be zero, so the test of γ =γ 2 reduces to a simple test of one coefficient bein snificantly different from zero. The model we then estimate is w = α + β lo y + κhhsize + δchildren

10 We want to test H 0 : δ = 0 vs. H A : δ 0. As we have assumed the residual to be normally distributed, we now that ˆδ (δ, seδ ). As we have to estimate the variance of the residual, it follows that under the null hypothesis, T = ˆδ se δ t (2247 4) We chose a 5% level of snificance. We then reject the null hypothesis if T >,96 (usin 8 derees of freedom to approximate 2243). The reression results are shown below: Constant Ltotexp children hhsize sma RSS R^ F(3,2243) = 6 [0.000]** lo-lielihood DW.05 no. of observations 2247 no. of parameters 4 mean(w_rice) var(w_rice) We find a value of the test statistic of T=3,4, which permits us to reject the null hypothesis at the 5% level of snificance. If the null were true, it would be a probability of 5% or less of ettin data that yields a test statistic of this manitude or above. Hence we can conclude that adults and children have snificantly different impacts on the demand behaviour. d) We want to test whether there is discrimination amon boys and irls. To do this, I will first study the effect of females and males in the different ae roups in the supplied data set. I then have a model of the form w i = α + β lo y i + γ females in + δ males in + ε i

11 where there are ae roups and γ and δ are the coefficients on the number of females and males in the roup. This reression ives the followin result: Constant Ltotexp f ft f5t f0t f5t f20t f35t f50t m mt m5t m0t m5t m20t m35t m50t sma RSS R^ F(7,2229) = [0.000]** lo-lielihood DW.07 no. of observations 2247 no. of parameters 8 mean(w_rice) var(w_rice) To test whether there are snificant differences, notice that we can rewrite the model as w i = α + β lo y i + θ females in + κ persons in + ε i.

12 Under the null hypothesis of γ = δ, this model would ive θ = 0, which we wish to test aainst an alternative hypothesis of γ δ. We can then run a reression with number of females and total number of persons in each roup, and the test of discrimination within this roup reduces to testin whether the coefficient on females is snificantly different from zero. This reression is iven below: Constant Ltotexp f ft f5t f0t f5t f20t f35t f50t c ct c5t c0t c5t c20t c35t c50t sma RSS R^ F(7,2229) = [0.000]** lo-lielihood DW.07 no. of observations 2247 no. of parameters 8 mean(w_rice) var(w_rice)

13 The variables startin with c denotes the number of people in the actual roup. We notice that except for the roups 5 to 9 and 25 to 49, we cannot reject the null hypothesis of no discrimination. Within these two roups, the coefficient on women is hher, so it seems they et more food. To ive an overall test of discrimination, we want to test a null hypothesis H 0 : γ = δ for all vs. the alternative H A : γ δ for at least one. To test this hypothesis, we consider a reression where the null is imposed, This is one where one the number of persons in each roup is included and the decomposition into sexes is excluded. Such a reression is reported below: Constant Ltotexp c ct c5t c0t c5t c20t c35t c50t sma RSS R^ F(9,2237) = [0.000]** lo-lielihood DW.04 no. of observations 2247 no. of parameters 0 mean(w_rice) var(w_rice) We now that iven the assumptions of the model, under the null hypothesis, the test statistic F = (RSS R RSS U ) /q F (q, ) RSS U / where q is the number of restrictions, here 8, the number of explanatory variables in the unconstrained reression, here 8, and RSS R and RSS U the RSSs from the restricted and

14 unrestricted reressions. At the 5% level, we reject the null hypothesis if we observe F above,94 (usin df in the denominator). Calculation yields F = ( ) / / (2247 8) = which is below the critical value. Hence we cannot reject the null hypothesis of no overall discrimination. e) Instead of looin at the demand for rice, we now want to study the demand for adult oods as this may ive a cleaner view of the total expenses on ids. Otherwise, the analysis is analoue to the one above. First, we study the effect of the number of females and males in each ae roup on demand: Constant Ltotexp f ft f5t f0t f5t f20t f35t f50t m mt m5t m0t m5t m20t m35t m50t

15 sma RSS R^ F(7,2229) = [0.000]** lo-lielihood DW.7 no. of observations 2247 no. of parameters 8 mean(w_adult) var(w_adult) o clear pattern emeres. To test whether there are snificant differences, aain we replace males by total number of persons in each roup and loo at whether the coefficient on females are snificant: Constant Ltotexp f ft f5t f0t f5t f20t f35t f50t c ct c5t c0t c5t c20t c35t c50t sma RSS

16 R^ F(7,2229) = [0.000]** lo-lielihood DW.7 no. of observations 2247 no. of parameters 8 mean(w_adult) var(w_adult) Aain, there are no snificant differences for the children as the t-values on females are below the critical value of,96 for all ae roups except 20 to 34. However, there are some sns of discrimination for adults, which may indicate that adult oods are mostly consumed by men. To test the hypothesis of overall discrimination, I aain run the constrained reression: Constant Ltotexp c ct c5t e c0t c5t c20t c35t e c50t sma RSS R^ F(9,2237) = [0.000]** lo-lielihood 3974 DW.7 no. of observations 2247 no. of parameters 0 mean(w_adult) var(w_adult) The F-test is exactly as above, and aain I reject the null if I observe a statistic above,94. ow I et F = ( ) / / (2247 8) =. 07 7,

17 which aain is under the critical value. Hence we cannot reject the null hypothesis of no discrimination. h) Finally, we want to study the effect of villae size. I now o bac to a simpler demoraphic specification where I only have adults and children, and I also study the demand for rice. I et: Constant Ltotexp children adults vilsize e e sma RSS R^ F(4,2242) = 24.3 [0.000]** lo-lielihood DW.06 no. of observations 2247 no. of parameters 5 mean(w_rice) var(w_rice) Villae size is seen to have a numerically small impact, but the coefficient is snificantly different from zero as the t-value is Hence people in larer villaes tend to have a smaller budet share on rice then people in smaller villaes, other thins equal. It seems strane that villae size in itself should have an impact on the demand for rice, so the most plausible explanation is that villae size is correlated with an omitted variable. We could imaine several, for instance that in larer villaes, the maret structure could be different, so prices differ or households demand other oods. It could also be that the linear specification of lo expenditure of adults and children is wron. If villae size is correlated with say the square of number of children, and number of children has a non-linear effect on the budet share for rice, this could explain the findin.