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Econometrics Paper
Minimize = ….we’ll come back to this later
For now, however, we can manually compute the residuals by hand. Fill in the table.


i 


70  95  
65  100  
90  120  
85  140  
110  160  
115  194  
120  265  
148  220  
155  236  
150  260  
verify this sum = 0 
The symbolshould have a subscript i to indicate it is not a constant, however, the equation editor does not depict this in a visually appealing manner.
Problem Set A (Fall 2016) You may work with anyone (fellow classmates, not an outside professional) but hand in your own paper. Hand in one, single document (please use the abbreviated answer sheet provided) that is stapled or bounded together in a very, professional manner. _____________________________________________________________ 1. Presented below are hypothetical data on weekly family consumption expenditure Y and weekly family income X. A. Obtain the sample regression function (with computed values) for this data using SAS (write the program in the SAS editor using a cards statement as we did in class). Provide the following: Using symbols, what does the population regression function (PRF) look like (refer to the text if needed)? _____________________________________________________________ Using symbols, what does the sample regression function (SRF) look like? _____________________________________________________________ Based on your SAS output, what is the SRF with the estimated values? ____________________________________________________________ B.
Obtain a correlation matrix for this data. What is the coefficient of correlation indicating about the direction and strength of the 2 variables? C. Now we will produce the sample regression estimates by hand. Compute the sample intercept b1 and sample slope b2 manually. Remind me to give you the formulas in lecture. To help, use the columns set up below to start: yi xi 70 95 65 100 90 120 85 140 110 160 115 194 120 265 148 220 155 236 150 260 D.
We discussed in class the nature of the residuals, ei . When we do Ordinary Least Squares regression (OLS), we choose b1 and b2 in such a way that the residuals are as small as possible1 . The way we do this is to make the residual sum of squares (RSS), ei 2 , as small as possible. In other words, we have a minimization problem!
Minimize ei 2 = y y i ^ 2 ….we’ll come back to this later!2 For now, however, we can manually compute the residuals by hand. Fill in the table. 1 Recall, the residuals measure differences between the actual and estimated y values, where ei = yi – y ^ . The estimated y ^ , or predicted y ^ , is what the regression model predicts for y. 2 The symbol y ^ should have a subscript i to indicate it is not a constant, however, the equation editor does not depict this in a visually appealing manner. yi xi y ^ i ei ei 2 70 80 65 100 90 120 95 140 110 160 115 180 120 200 140 220 155 240 150 260 verify this sum = 0 E. From the calculations in the table, what is the value of RSS? ____________________________________________________________ For the SRF being studied here, the degrees of freedom are n – k. The mean squared error (MSE) is n k RSS , and this is also known as the variance for the whole model. F. What is the value of the mean squared error term (hint: SAS helps you here too)? ____________________________________________________________ There is also something called the standard error of the estimate, which is a measure of how scattered the original data points are around the regression line being estimated. This is also known as the root mean squared error term (RMSE), and is equal to n k RSS . G. What is the value of the root mean squared error term? ____________________________________________________________ Ok, now that we know something about the residual sum of squares, a.k.a., RSS, a.k.a., y y i ^ 2 , a.k.a., ei 2 , we can talk about the total sum of squares (TSS) and the explained sum of squares (ESS). All of these terms are important in determining the golden statistic known as the coefficient of determination ( r 2 ). The TSS can be described as the total variation of the actual y values about their sample mean. We can express it algebraically as: y y i __ 2 . The ESS can be described as the variation of the estimated y values about their mean. We can express it algebraically as y y ^ __ 2 . Finally, the TSS = ESS + RSS. Take a moment to locate the values for each of these on the SAS output. Caution: SAS reports the ESS as the Sum of Squares Model, the RSS as the Sum of Squares Error, and the TSS as the Sum of Squares Corrected Total. H. ESS = __________ I. RSS = __________ J. TSS = __________ K. Generate the diagram for this example illustrating the TSS, ESS, and RSS as we did in lecture. You may pick one data point as a reference in the same way we did in lecture. 2. Consider the output below. Suppose we infer a relationship between the heights of daughters (Xi) and the heights of their mothers (Yi). Using the ouput, and it looks some parts are missing because you dog chewed it, determine the values for the following: RSS MSE, or S2 R 2 b2 The expression for the SRF You also know that Xbar=63.78, Ybar= 63.75 3. Using the data in the excel file called gpa (the data is reproduced below for your convenience), estimate a model in SAS that uses ACT scores to predict GPA. student gpa act 1 2.8 21 2 3.4 24 3 3 26 4 3.5 27 5 3.6 29 6 3 25 7 2.7 25 8 3.7 30 A. Write out the full equation with the estimate values generated by SAS. Does the intercept coefficient have a useful, intuitive interpretation here? B. How much higher is GPA predicted to be if the ACT score is increased by 5 points? C. Notice in your SAS code that you probably included syntax that looked like this: proc reg ; model gpa=act ; What I’d like you to do is add the line: proc reg ; model gpa=act ; output out=new r=res p=pred ; The output line produces the residuals (called res) and predicted values for GPA (called pred) and puts them into a temporary data set called new. Now, we want to work with the residuals. Create another data set that references “new” and plot the squared residuals against the independent variable act (put the squared residuals on vertical). Describe what you see? A pattern? Note: We will go over this process in class with the cigarette demand curve example. 4. Read Gujarati’s discussion on summation operators, and then expand the following summation operators into expanded algebraic expressions as far as possible: a. b. c. For part d., do the reverse and simplify the expanded algebraic expression into a summation operator: d. 5. Show, using summation algebra, that Σei = 0 6. Answer the following: Consider the data for the quantity demanded of the mineral plastonia and the relative price of plastonia. Interpret the coefficient for b2. log(qd plastonia) = b1 + b2log(pplastonia) + ei b. Consider that hourly wage and years of education are related by the functional form below. Interpret the coefficient for b2. log(wage) = b1 + b2(education) + ei 7. Using the data in the excel file called hprice2play_fall 2014 (the data is reproduced below for your convenience) to estimate the regression of the following functional form: log(price) = b1 + b2log(nox) + b3(rooms) + ei, where price=price of a house in community i, nox=a proxy for pollution that is the nitrous oxide in the air over community i, and rooms is the # of rooms in the house in community i. Also, log is the natural log. a. Interpret the coefficients on b2 and b3. b2 b3 b. Perform an Ftest. Show the null and alternative hypothesis, using symbols, and present your decision and conclusion. price crime nox rooms dist radial proptax stratio lowstat 24000 0.006 5.38 6.57 4.09 1 29.6 15.3 4.98 21599 0.027 4.69 6.42 4.97 2 24.2 17.8 9.14 34700 0.027 4.69 7.18 4.97 2 24.2 17.8 4.03 33400 0.032 4.58 7 6.06 3 22.2 18.7 2.94 36199 0.069 4.58 7.15 6.06 3 22.2 18.7 5.33 28701 0.03 4.58 6.43 6.06 3 22.2 18.7 5.21 22900 0.088 5.24 6.01 5.56 5 31.1 15.2 12.43 27100 0.145 5.24 6.17 5.95 5 31.1 15.2 19.15 16500 0.211 5.24 5.63 6.08 5 31.1 15.2 29.93 18900 0.17 5.24 6 6.59 5 31.1 15.2 17.1 15000 0.225 5.24 6.38 6.35 5 31.1 15.2 20.45 18900 0.117 5.24 6.01 6.23 5 31.1 15.2 13.27 21700 0.094 5.24 5.89 5.45 5 31.1 15.2 15.71 20400 0.63 5.38 5.95 4.71 4 30.7 21 8.26 18200 0.638 5.38 6.1 4.46 4 30.7 21 10.26 19900 0.627 5.38 5.83 4.5 4 30.7 21 8.47 23100 1.054 5.38 5.93 4.5 4 30.7 21 6.58 17500 0.784 5.38 5.99 4.26 4 30.7 21 14.67 20200 0.803 5.38 5.46 3.8 4 30.7 21 11.69 18200 0.726 5.38 5.73 3.8 4 30.7 21 11.28 13600 1.252 5.38 5.57 3.8 4 30.7 21 21.02 19600 0.852 5.38 5.96 4.01 4 30.7 21 13.83 15200 1.232 5.38 6.14 3.98 4 30.7 21 18.72 14500 0.988 5.38 5.81 4.1 4 30.7 21 19.88 15600 0.75 5.38 5.92 4.4 4 30.7 21 16.3 13900 0.841 5.38 5.6 4.45 4 30.7 21 16.51 16600 0.672 5.38 5.81 4.68 4 30.7 21 14.81 14800 0.956 5.38 6.05 4.45 4 30.7 21 17.28 18400 0.773 5.38 6.49 4.45 4 30.7 21 12.8 8. More practice running regressions. Metro area Property crimes /100, yi Unemployment rate xi Santa Barbara 2528 .076 Honolulu 3679 .038 Fort Smith 5861 .070 Hattiesburg 5841 .063 Billings 4360 .042 Rome 6298 .103 Napa 2554 .085 Reno 3814 .110 Davenport 4914 .078 Chico 2776 .124 SebastianVero beach 3216 .119 Texarkana 6911 .064 Midland 3695 .045 St. George 2043 .066 a. Using the data above, produce a scatterplot (put crime on vertical) and comment on what you see. b. Produce a coefficient of correlation and comment on the direction and strength between the two variables. c. Estimate a regression that predicts property crime rates using unemployment rates for the metro areas in the year 2009. Write the equation below with the estimated values. d. As we did in the gpa example, produce a scatterplot of the residuals (on the vertical) against the independent variable. Comment on what you see based on our discussions in class. e. Describe if there’s any possibility that the causal direction could be flipped. Provide an explanation for your reasoning.
student  gpa  act 
1  2.8  21 
2  3.4  24 
3  3  26 
4  3.5  27 
5  3.6  29 
6  3  25 
7  2.7  25 
8  3.7  30 
price  crime  nox  rooms  dist  radial  proptax  stratio  lowstat 
24000  0.006  5.38  6.57  4.09  1  29.6  15.3  4.98 
21599  0.027  4.69  6.42  4.97  2  24.2  17.8  9.14 
34700  0.027  4.69  7.18  4.97  2  24.2  17.8  4.03 
33400  0.032  4.58  7  6.06  3  22.2  18.7  2.94 
36199  0.069  4.58  7.15  6.06  3  22.2  18.7  5.33 
28701  0.03  4.58  6.43  6.06  3  22.2  18.7  5.21 
22900  0.088  5.24  6.01  5.56  5  31.1  15.2  12.43 
27100  0.145  5.24  6.17  5.95  5  31.1  15.2  19.15 
16500  0.211  5.24  5.63  6.08  5  31.1  15.2  29.93 
18900  0.17  5.24  6  6.59  5  31.1  15.2  17.1 
15000  0.225  5.24  6.38  6.35  5  31.1  15.2  20.45 
18900  0.117  5.24  6.01  6.23  5  31.1  15.2  13.27 
21700  0.094  5.24  5.89  5.45  5  31.1  15.2  15.71 
20400  0.63  5.38  5.95  4.71  4  30.7  21  8.26 
18200  0.638  5.38  6.1  4.46  4  30.7  21  10.26 
19900  0.627  5.38  5.83  4.5  4  30.7  21  8.47 
23100  1.054  5.38  5.93  4.5  4  30.7  21  6.58 
17500  0.784  5.38  5.99  4.26  4  30.7  21  14.67 
20200  0.803  5.38  5.46  3.8  4  30.7  21  11.69 
18200  0.726  5.38  5.73  3.8  4  30.7  21  11.28 
13600  1.252  5.38  5.57  3.8  4  30.7  21  21.02 
19600  0.852  5.38  5.96  4.01  4  30.7  21  13.83 
15200  1.232  5.38  6.14  3.98  4  30.7  21  18.72 
14500  0.988  5.38  5.81  4.1  4  30.7  21  19.88 
15600  0.75  5.38  5.92  4.4  4  30.7  21  16.3 
13900  0.841  5.38  5.6  4.45  4  30.7  21  16.51 
16600  0.672  5.38  5.81  4.68  4  30.7  21  14.81 
14800  0.956  5.38  6.05  4.45  4  30.7  21  17.28 
18400  0.773  5.38  6.49  4.45  4  30.7  21  12.8 
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