3305AFE - Applied Econometrics - Ordinary Least Squares - Economics Assignment Help

Question 1.
Regression models are a workhorse method of statistical and/or econometric analysis. These models can take the form y=?0+?1x+e, where y is the dependent variable and x the independent variable.
These equations are usually fit to data using a technique known as “Ordinary Least Squares”, or OLS. Briefly explain the mechanics and intuition behind this method. Why is a model estimated using OLS considered a “line of best fit” for a given data set?
OLS estimates the parameters in a regression model by minimizing the sum of the squared residuals. It estimates the unknown parameters. This method draws a line through the data points which minimizes the squared differences between the observed values and corresponding values
Using a couple of sentences in each case, explain the meanings behind the following terms in the context of the above regression model.
R2R Squared is a statistical measure that is used of how close the data points are to the regression line which is also known as the coefficient of determination. It is used in regression analysis to determine the goodness of fit for linear regression models.

In relation to the OLS regression model this symbol represents the coefficient estimator
Type equation here.eThis is the error term when it comes to predicting Y given the value of X. Within a linear regression model tracking a stock’s price over time the error term is the difference between the expected price at a particular time and price it was observed. It also represents the margin of error within a statistics model. This refers to the sum of deviations within the regression line this provides an explanation for the t theoretical values and the actual observed results
Regression models often employ the assumptions Vare=?2 and covei,ej=0. Explain the meanings of these two terms. You may like to illustrate these assumptions (or their violations) by providing graphs such as scatter plots. What implications do they have for (i) the standard errors from the model, and (ii) the Gauss-Markov Theorem?
These two terms represent estimating the variance and covariance of the error term. The variance of the random error term is the one parameter of the simple linear regression model that remains to be estimated

Question 2.
Finance professionals are often interested in minimizing risk in their portfolios by investing in assets that react differently under varying market conditions. The idea is that by buying some securities that are positively correlated with broader market movements, and some that are negatively associated with the market, the combined risk exposure will be reduced.
This type of risk can be measured for a share using the market beta - a parameter from a regression model designed to measure the association between the return on the asset and the overall market performance. Market betas can be calculated using the following equation:
ys=?0+?1xm+e where ys is the return on the asset, and xm is the market return? A share with a high beta will move strongly with the market, while a beta closer to zero will be less sensitive to market fluctuations. Shares with negative betas will move in the opposite direction to the broader market.
Table 1 below gives an estimate for a US firm that manufactures textiles.
Table 1. Asset Returns and Market Returns - Textiles
Dependent Variable: ASSET RETURN Method: Least Squares Sample: 1 32 Included observations: 32 Variable Coefficient Std. Error t-Statistic Prob.  
C 0.332108 0.145082 2.289111 0.0293
MARKET RETURN 0.421084 0.173615 2.425387 0.0215
R-squared 0.163938     Mean dependent var 0.472250
Adjusted R-squared 0.136069     S.D. dependent var 0.809924
S.E. of regression 0.752807     Akaike info criterion 2.330447
Sum squared resid17.00157     Schwarz criterion 2.422055
Log likelihood -35.28715     Hannan-Quinn criter. 2.360812
F-statistic 5.882501     Durbin-Watson stat 2.555932
Prob(F-statistic) 0.021521
Do your asset returns move with the market, against the market, or are uncorrelated with the market? Briefly explain.
Calculate a 90% confidence interval for the market beta (?1) using information drawn from the output. Show all working and provide an interpretation for your result.
Test the null hypothesis that there is no link between the returns on your specific asset (ys) and the return on the market (xm) at ?=5%. Give the null and alternative hypotheses, a test (t) statistic, critical value, p-value and a conclusion.

Question 3.
A criminologist is interested in the social and economic factors that contribute to violent crime. To study this issue, she takes data on incidents of violence per 100,000 people per year, and regresses this against measures of poverty, education, income and unemployment. All variables are collected at the geographical level. The model she employs is
yVC=?0+?1xP+?2xE+?3xI+?4xU+ewhere xP, xE, xI and xU are the poverty, education, income and unemployment variables respectively. Poverty is measured in percentage points, average education in years, income in thousands of dollars per year, and unemployment in percent.
An output of her model is given below.
Table 2. Economic Determinants of Violent Crime
Dependent Variable: VIOLENT CRIME (Per 100,000) Method: Least Squares Sample: 1 65 Included observations: 65 Variable Coefficient Std. Error t-Statistic Prob.  
C 257.5038 33.12690 7.773253 0.0000
P 2.674599 1.095255 2.441988 0.0176
E 0.680625 1.925346 0.353508 0.7249
I -1.015186 0.365762 -2.775538 0.0073
U 1.657919 1.689903 0.981074 0.3305
R-squared 0.194721     Mean dependent var 247.1954
Adjusted R-squared 0.141035     S.D. dependent var 33.41308
S.E. of regression 30.96736     Akaike info criterion 9.777548
Sum squared resid57538.64     Schwarz criterion 9.944809
Log likelihood -312.7703     Hannan-Quinn criter. 9.843543
F-statistic 3.627078     Durbin-Watson stat 1.992742
Prob(F-statistic) 0.010317
Provide an interpretation of the parameter ?1 (the coefficient on poverty). How does this interpretation differ from one obtained from a model where education, income and unemployment are excluded (i.e. the model yVC=?0+?1xP+e)?
The coefficient would be different considering the effect that education, income and unemployment would have on poverty in general and the slope of the linear regression would also be affected
Which variables appear to be the most significant determinants of violent crime? Which variable is the least significant? Provide an interpretation of this model that could be useful for a policy maker who is trying to lower violent crime in their district.
What fraction of the overall variation in yVC is explained by the covariates in the model? Make some suggestions to the criminologist as to how the model fit could be improved.
Suppose a region has a poverty rate of 8%, an average educational attainment of 11.2 years, an average income of $42 (000) per year, and an unemployment rate of 7%. Provide a prediction of the rate of violent crime for this geographical area.

Question 4.
This question uses the information from Question 3.
Alongside the model estimated in Table 2, the criminologist also estimates an equation of the form yVC=?0+e and the output is given below.
Table 3. Economic Determinants of Violent Crime – Null Model
Dependent Variable: VIOLENT CRIME (Per 100,000) Method: Least Squares Sample: 1 65 Included observations: 65 Variable Coefficient Std. Error t-Statistic Prob.  
C 247.1954 4.144383 59.64588 0.0000
R-squared 0.000000     Mean dependent var 247.1954
Adjusted R-squared 0.000000     S.D. dependent var 33.41308
S.E. of regression 33.41308     Akaike info criterion 9.871037
Sum squared resid71451.79     Schwarz criterion 9.904489
Log likelihood -319.8087     Hannan-Quinn criter. 9.884236
Durbin-Watson stat 2.014774
Using information drawn from Tables 2 and 3, perform an F-test for the overall significance of the model presented in Table 2. Give the unrestricted and restricted models, the null and alternative hypotheses, values for F-calc and F-crit, and a conclusion.
Briefly explain the intuition behind the F-test you performed above. If SSER and SSEU turn out to be very similar, what would this imply about the null hypothesis in an F test?
The criminologist is interested in assessing the functional form of the model depicted in Table 2. She performs the RESET test and the output is provided overleaf.
Using the output in Table 4, conduct the RESET test. Give the null and alternative hypotheses, the F-statistic, P-value and a conclusion. Does the model she has used have the correct functional form?
Briefly give an explanation of how the RESET test works. Provide some intuition around the role of the additional non-linear terms added to the equation in assessing the specification.
Table 4. RESET Test – Violent Crime Model
Ramsey RESET Test Equation: UNTITLED Specification: VC C P E I U Omitted Variables: Powers of fitted values from 2 to 3
Value df Probability F-statistic  0.700956 (2, 58)  0.5003 Likelihood ratio  1.552421  2  0.4601 F-test summary: Sum of Sq. df Mean Squares Test SSR  1357.938  2  678.9689 Restricted SSR  57538.64  60  958.9774 Unrestricted SSR  56180.71  58  968.6329 LR test summary: Value Restricted LogL-312.7703 Unrestricted LogL-311.9941 Unrestricted Test Equation: Dependent Variable: VC Method: Least Squares Sample: 1 65 Included observations: 65 Variable Coefficient Std. Error t-Statistic Prob.  
C -24781.88 21216.71 -1.168036 0.2476
P -376.8784 321.2959 -1.172994 0.2456
E -95.69033 81.59221 -1.172788 0.2457
I 143.0923 121.9890 1.172993 0.2456
U -233.6525 199.2176 -1.172851 0.2457
FITTED^2 0.581268 0.493189 1.178592 0.2434
FITTED^3 -0.000791 0.000673 -1.174969 0.2448
R-squared 0.213726     Mean dependent var 247.1954
Adjusted R-squared 0.132387     S.D. dependent var 33.41308
S.E. of regression 31.12287     Akaike info criterion 9.815203
Sum squared resid56180.71     Schwarz criterion 10.04937
Log likelihood -311.9941     Hannan-Quinn criter. 9.907596
F-statistic 2.627601     Durbin-Watson stat 2.023006
Prob(F-statistic) 0.025358

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