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THE UNIVERSITY OF HULLThe Business School
赫尔大学的商学院
Level 6 Examination
六年级考试
January 2011
Empirical Economics
Date : 26 January 2011
日期：2011.01.26
Time : 9.30amTime
Allowed : 2 hours
Answer ALL questions from section Aand ANY TWO questions from section B
Marks are indicated at the end of each subquestion in [ ]
回答所有的问题，从部分A和部分B两题
Do not open or turn over this exam paper, or start to writeanything http://www.ukassignment.org/essaygs until told to by the Invigilator. Starting to writebefore permitted to do so may be seen as an attempt to useUnfair Means.
不要打开或关闭这项考试，或直到被告知监考员，开始写东西。开始写之前允许这样做可能被视为试图使用不正当手段。
Section A: Compulsory Question
A：必答题
Q1. Consider a regression of the supply of a product (Yi) on its own prices (Xi) as follows
Yi=1+2Xi+ eii = 1 :::N
where eiis a randomly distributed error term for observation i.
a. Discuss the economic theory behind this model and list the OLS assumptions on errorterms ei. [5]
b. Derive the normal equations and the OLS estimatorsb1andb2: [10]
b1andb2: [10]c. A supplier observed the data on quantities and prices as given in Table 1 below. Whatare the OLS estimatesb1andb2implied by these data?
[15]Table 1: Data on Quantities and PricesQuantities (Yi) 5 3 7 9 6 4Prices (Xi) 7 5 8 10 8 5d. What are the variances of eiand Yi?
d. What are the variances of eiand Yi?
f. Determine the overall signi…cance of this model by F -test at a 5 percent level of sig-ni…cance. [Critical value of F for df(1,4) =7.71] [10]
g. What are the variances and standard errors ofb1andb2?
b1andb2? [10]h. Compute t-statistics and determine whether parametersb1andb2are statisticallysigni…cant at a 5 percent level of signi…cance. [Critical value of t for a 5% signi…cance for4 degrees of freedom is 2.776 (i.e. tcrit;0:05;4= 2:776) ] [10]
i. What is the prediction of Y when X is 15?
j. What is the elasticity of supply evaluated at the mean values of Yiand Xi? [5]
k. Reformulate the model to include the price of a related product in the model. What willhappen to this estimation if the two prices are exactly correlated ? [5]
你会如何决定是否供应此产品，即夏，秋，冬，春四季变化？
l. How would you decide whether the supply of this product varies by four seasons, namelySummer, Autumn, Winter and Spring? [5]
Section B: Answer Any Two Questions
想出一个对某种产品的需求的多元回归模型，其中质量取决于产品的价格（X1 ）和个人的收入（X2）：
Q2. Consider a multiple regression model for the demand for a certain product, where thequantity (Yi) depends on price of the product (X1;i) and the income of individuals (X2;i) as:
Yi=0+1X1;i+2X2;i+ eii = 1 :::N
a. Provide a reasonable theoretical basis for the speci…cation of a demand function of thissort (e.g. consumer optimisation with Cobb-Douglas utility function). [10]
b. Write normal equations to derive the OLS estimators of0;1and2. [10]
c. Derive estimators of0;1and2: (hint: use deviation form and/or the Cramer’s rule).[10]
d. Write the expression for the variance of the error term. [14]
e. What is the covariance matrix of0;1and2? Write an expression for the variancesof0;1and2: [14]
f. Determine the test statistics for signi…cance of each of0;1and2and to test thegeneral hypothesis that0= 0;1= 0;and2= 0: [14]
g. What would happen to the OLS estimators derived above if X2;i= 5X1;ifor all i ? Whatcan be done to improve the validity of model in such a situation? [14]
h. Assume that instead of estimating Yi=0+1X1;i+2X2;i+eia researcher estimatedYi=0+1X1;i+ ei. How could you determine the statistical signi…cance of the revisedmodel? [14]
Q3. Consider the Gauss-Markov theorems for a linear regression of the demand for a normalgood (Yi) on its own prices (Xi) asYi=1+2Xi+ eii = 1 :::N
where eiis a randomly distributed white noise.
a. Prove that the OLS estimators of1and2are linear in Yi. [16]
b. Show that the OLS estimators of1and2are unbiased. [16]c. Verify that the OLS estimators of1and2have minimum variance and are consistent.[20]d. Discuss the nature of the problem which arises when E(Xiei)6= 0 [16]
e. Describe what happens to the OLS estimators of1and2when var(ei) =2i. [16]
f. Analyse what happens to the OLS estimators of1and2when E(eiej)6= 0: . [16][26163] [Continued...]3Q4. A piece of research using the UK data aims to study whether the level of unemployment(Yi) relates to the level of the bene…t-claimant count (X1;i), numbers of stoppages (X2;i) andwork hours (X3;i) taking data from the Labour Force Survey for 19 years (N = 19) asfollowing:Yi=0+1X1;i+2X2;i+3X3;i+ "ii = 1 :::N
f. Consider estimates put forward by a researcher on the aggregate supply function for UKusing quarterly data from 1960:1 to 2006:1 provided in tables 3 to 6 above. Criticallyjustify which ones of these results stands to the scrutiny of Durbin-Watsonstatistics and other statistical tests. Is there evidence for an aggregatesupply function? [20]Note: For K =2 the relevant Durbin-Watson dLand dUstatistic at a 1 percent and 5 percentlevel of signi…cance are given Table 7 below.Table 7: Relevant Part of Durbin-Watson Table1% 5%dLdUdLdUN=150 1.598 1.651 1.706 1.760N=200 1.653 1.693 1.748 1.789
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