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Coefficients Standard Error t Stat p-Value Intercept –170.8001942 123.872687 –1.378836597 0.210392 NI 0.095522952 0.011484578 8.317497956 7.1E-05 Chapter Problems Computational Problems 2.
X Y (X – X̄)2 (Y – Ῡ)2 (X – X̄) (Y – Ῡ) Sums 146.00 1105.00 60.18 4890.73 382.64 n = 11 SSX SSY SSXY
Means 13.273 100.455
b1 = 6.358 b0 = 16.067
B. (i) 𝑌� = 73.289; (ii) 𝑌� = 16.067 + 6.358(18) = 130.85. C. Income increases by $6,358. 4. It makes no difference which variable you identify as X and which as Y if you want only to calculate the r2 using Equation (2.6.3). r2 = 0.0012 indicating a very weak relationship. 6.
Debt GDP Population Debt/GDP Debt/Pop 1 1.56 34 0.641026 0.029412
B. 73.6% of any increase in income will go toward consumption. C. You will increase consumption by $736. D. r2 = 0.0599; apparently, many other economic forces contribute to changes in consumption patterns. E.
H0: ρ = 0 Sr = 0.10978336
Critical t or Z = 1.95996398 t or Ztest = 2.22959981
REJ p-value for Z = 0.02577402 p-values for t = 0.02865043
CHAPTER 4 Section 4.1 2. Even at the highest level of significance of 0.01, we find t.01,19 = 2.861 < either of the variables’ t-values. Thus, reject the nulls. Both are statistically significant at any common alpha value. =TDIST(3.66,19,2) = 0.0017. If you use the t-table the alpha value is less than 1%. =TDIST(5.26,19,2) = 0.00. 4.
Coefficients Standard
Error t Stat p-Value Intercept 5.382573 6.975536 0.771636 0.458176 X1 19.08475 0.907834 21.02229 1.32E-09 X2 –4.7808 0.96402 –4.95924 0.000571
Multiple R 0.988919 R Square 0.977962 Adjusted R Square 0.973554 Standard Error 7.495381 Observations 13
H0: b1 = 0 Sb = 0.32992006
Critical t or Z = 1.95996398 t or Ztest = 2.23084344
REJ p-value for Z = 0.0256915 p-value for t = 0.02865043
Section 4.2 2. Because it increases the standard error of the regression coefficient thereby decreasing the absolute value of the t-statistic used in the hypothesis test and moving it into the do not reject region. 4. ttest = 3.997; t.05,10 = 2.2281; reject null. Section 4.3 4. ANOVA
Intercept –2.224475192 6.155752718 –0.36137 0.71993867 TEMPERATURE 0.427337249 0.078528794 5.441791 3.87409E-06 KC –2.897785932 0.836088219 –3.46589 0.001384508 CHICAGO –2.96855398 0.702321909 –4.22677 0.000154734
CHAPTER 6 Section 6.1 4. Response Information Variable Value Count BANKRUPTCY 1 24 (Event) 0 18 Total 42 Logistic Regression Table
-6
-5
-4
-3
-2
-1
0
1
2
3
20 22 24 26 28 30 32 34 36
Residuals
Odds 95% CI Predictor Coef SE Coef Z P Ratio Lower Upper Constant –29.1676 11.2547 –2.59 0.010 INCOME 0.353613 0.161580 2.19 0.029 1.42 1.04 1.95 DEBT 0.0275068 0.0155788 1.77 0.077 1.03 1.00 1.06 Log-Likelihood = –14.748 Test that all slopes are zero: G = 27.869, DF = 2, P-Value = 0.000
6. χ2 = 49.3 – 42.7 = 6.6; using Excel =chiinv(.01,4) yields critical χ2 of 13.2767 > 6.6; Do Not Reject. The p-value is =chidist(6.6,4) yields 0.158598. Section 6.2 4. A few of the observations appear as:
Yes, they should visit this firm. 4. 42.7 = 13.7 + 29. 13.7 > 11.344. Yes, adding the RHS variables improves the model.
CHAPTER 7 Section 7.1 2. Increasing the sample size will do nothing to mitigate the ill-effects of heteroscedasticity. Section 7.2 2. The regression equation is LnResSq = – 1.64 + 0.107 LnPop Predictor Coef SE Coef T P Constant –1.638 1.862 –0.88 0.395 LnPop 0.1069 0.4558 0.23 0.818
The coefficient of log population is not significant. No evidence of heteroscedasticity. 4. The regression equation is ResSq = – 7.4 + 1.76 BR – 0.0541 Pop – 0.34 InfMor – 0.085 BRSQ + 0.000005 PopSq – 0.0069 InfMorSq + 0.00352 BRPop + 0.044 BRInfMor + 0.00022 PopInfMor Predictor Coef SE Coef T P Constant –7.43 12.09 –0.61 0.566 BR 1.758 2.490 0.71 0.512 Pop –0.05408 0.08948 –0.60 0.572 InfMor –0.339 1.574 –0.22 0.838 BRSQ –0.0846 0.1244 –0.68 0.527 PopSq 0.00000464 0.00002991 0.16 0.883 InfMorSq –0.00691 0.02127 –0.33 0.758 BRPop 0.003519 0.006504 0.54 0.612
nR2 = (15)(0.548) = 8.22; =chidist(.01,7) = 18.4753 > 8.22. Do Not Reject null of NO heteroscedasticity. 6. Predictor Coef SE Coef T P Constant –9.258 2.210 –4.19 0.000 LnBB 2.5228 0.5981 4.22 0.000
The p-value for LnBB is significant. Heteroscedasticity can be assumed to exist. 8. The regression equation is RedSq = 23.0 + 0.82 Foreign Investment – 2.21 Bank Balances + 0.0000 ForInvSq + 0.0755 BBSQ – 0.0529 FIBB
nR2 = (26)(0.71) = 18.46; χ2 with alpha of .01 and d.f. = 5 is 15.08 < 18.46; Reject null of no heteroscedasticity; conclude heteroscedasticity exists. Section 7.3 2.
𝜈𝑖2 =𝜀𝑖2
𝜎𝑖2
𝐸(𝜈𝑖2) = 𝐸 �𝜀𝑖2
𝜎𝑖2�
= �1𝜎𝑖2�𝐸(𝜀𝑖2)
=1𝜎𝑖2
(𝜎𝑖2) = 1
4. It says that the population variances at the different X-values have a constant proportionality with the X-values squared. 6. The regression equation is BT/BB = 3.49 1/BB + 0.0324 BB/BB + 1.01 FI/BB
Multiply through by BB and we get BT = 3.49 + 0.0324BB + 1.01FI. Chapter Problems Computational Problems 2. The regression equation is lnAmt = 1.97 + 0.476 lnCost + 0.600 lnPop
Predictor Coef SE Coef T P Constant 1.969 2.816 0.70 0.492 lnCost 0.4759 0.4843 0.98 0.336 lnPop 0.6000 0.1285 4.67 0.000
The regression equation is Utility = 62.9 + 0.542 Wager Predictor Coef SE Coef T P Constant 62.873 1.529 41.11 0.000 Wager 0.54224 0.05447 9.95 0.000 S = 3.79491 R–Sq = 81.8% R–Sq(adj) = 81.0% The regression equation is lnUtility = 3.94 + 0.131 lnWager Predictor Coef SE Coef T P Constant 3.94398 0.00780 505.76 0.000 lnWager 0.131357 0.002566 51.18 0.000 S = 0.0118087 R–Sq = 99.2% R–Sq(adj) = 99.1%
43210
4.5
4.4
4.3
4.2
4.1
4.0
3.9
lnWager
lnUt
ility
Scatterplot of lnUtility vs lnWager
Chapter Problems
Computational Problems 2. Vertex = –b/2a = –15/–5 = 3 units of Q. At Q = 3, TR = 22.5. At both Q = 2 and Q = 4, TR = 20.
0 5 100
10
20
30
f x( )
x 4. TC = 89.96Q + 0.0179Q3 – 2.294Q2.
6. Regression Statistics
Multiple R 0.121449 R Square 0.01475 Adjusted R Square –0.02466 Standard Error 18.4823 Observations 27
Coefficients Standard
Error t Stat p-Value Intercept 28.244442 6.999174998 4.035395888 0.000452369 MILES 0.055875458 0.091333306 0.611775271 0.546210348
8. For the linear model R2 = 0.56 and Se = 9.9:
Coefficients Standard Error
t Stat p-Value
Intercept 28.92694444 7.214133 4.009760352 0.005125869 Q –1.91725 0.640993 –2.991064464 0.020195889 The residuals show a pattern. They violate the OLS assumption that error terms are random.
Not sure about the residuals. They may show a pattern! Can’t compare R2 values Ɛ = 1.5; an elastic demand curve. 10. The –0.8879 says for every one unit Q goes up, P must gown down by 0.8879 of its units. The coefficient for lnP = ln2.3 – 3.5lnQ says for every 1% Q goes up, P must go down by 3.5%, 12. At L = 10.1 (a 1% increase) Q = 186.131; a 0.6% increase. 14. At I = 10, lnC = 2.58 and C = 13.2. At I = 11, lnC = 2.638, C = 13.98. This is a 5.9% increase. (Difference is due to rounding.) 16.
01234567
0 20 40 60 80
Y
Y
Regression Statistics
Multiple R 0.756417455 R Square 0.572167366 Adjusted R Square 0.541607892 Standard Error 0.144243539 Observations 31
Coefficients Standard Error t Stat p-Value Intercept 5.016508896 0.008940949 561.0712156 4.46361E-60 1/X 2.240027339 0.076395157 29.32158836 4.17982E-23 Notice how the graph does not below the floor equal to the coefficient of the reciprocal model of 5.0165.
Regression Statistics
Multiple R 0.839634736 R Square 0.70498649 Adjusted R Square
Coefficients Standard Error t Stat p-Value Intercept 5.810332486 0.078788667 73.74578973 1.49616E-34 lnx –0.206027353 0.024748931 –8.324697172 3.54753E-09 18. (1258)(2.718–0.105) = 1133 > 1000. No down grade. 20. FV = 43,000(e0.055*6) = $59,811. Closer! 22. Currently, costs are 100(3) + 50(4) = 500. Revenues are 123(3.75) = 461 for a loss of 39. If more inputs are added to the business costs are 100(6) + 50(8) = 1000 and revenues are 984(3.2) = 3149 for a profit of 2149. 24.
GM = �6242
5 – 1 = 8%. In 2013 costs will be 62(1.08) = 66.96. In 2014 they will be
a. Dependent Variable: Q 4. Durbin–Wu–Housman test for Police in the equation for Crimes. Obtain the errors Police as a function of all exogenous variables Students, Poverty and taxes. Regress the structural equation for Crimes on the explanatory variables and the error values from the first regression run. The errors are significant. They are offering explanatory power to Crimes which means they are NOT random and are a source of bias.
Durbin–Wu–Housman test for Police in the equation for Police. Regress Crimes on all exogenous variables and use the errors to estimate the structural equation for Police. The error terms are not significant.
CHAPTER 11 Section 11.1 2. Using Year: The regression equation is TONS = – 1089 + 0.550 YEAR Predictor Coef SE Coef T P Constant –1089.5 109.5 –9.95 0.000 YEAR 0.55000 0.05455 10.08 0.000 S = 0.422577 R–Sq = 93.6% R–Sq(adj) = 92.6%
Using Time Periods 1 through 9: The regression equation is TONS = 12.2 + 0.550 TIME Predictor Coef SE Coef T P Constant 12.1500 0.3070 39.58 0.000 TIME 0.55000 0.05455 10.08 0.000
Forecast for 2016 = 19.3. 4. After creating a column for time periods 1 through 30, regress Time on Volume:
Regression Statistics Multiple R 0.348673126 R Square 0.121572949
Intercept 222845739.6 15377708 14.49147982 1.54E-14 191345933.9 Time Period –1705163.784 866206.5 –1.968541796 0.058978 –3479507.429
For Nov 16: 222845739.6 – 1705163.784(45) = –54447762.* *The data were collected during a period of world–wide economic instability and turmoil. Forecasts based on time trends were extremely tenuous. Reliance on them and the use of R2 values and standard errors was quite hazardous at best. Obviously, a negative value for volume is impossible. Furthermore the “45” used in the forecast included weekends when the market did not trade. Perhaps only trading days of 11 should be added on to the 30 already found in the data set making it 41 instead of 45. Section 11.2
2. Regression Analysis: TONS versus LagTons1: The regression equation is TONS = 3.09 + 0.830 LagTons1 8 cases used, 1 cases contain missing values Predictor Coef SE Coef T P Constant 3.091 2.593 1.19 0.278 LagTons1 0.8304 0.1768 4.70 0.003 S = 0.648304 R–Sq = 78.6% R–Sq(adj) = 75.1% Analysis of Variance Source DF SS MS F P Regression 1 9.2732 9.2732 22.06 0.003 Residual Error 6 2.5218 0.4203
Total 7 11.7950
4.
Regression analysis: Volume versus VolLag1: The regression equation is Volume = 1.24E+08 + 0.365 VolLag1 29 cases used, 1 cases contain missing values Predictor Coef SE Coef T P Constant 124083240 35817145 3.46 0.002 VolLag1 0.3652 0.1788 2.04 0.051 S = 41266719 R–Sq = 13.4% R–Sq(adj) = 10.2% Analysis of Variance Source DF SS MS F P Regression 1 7.10285E+15 7.10285E+15 4.17 0.051 Residual Error 27 4.59794E+16 1.70294E+15 Total 28 5.30823E+16
Section 11.3 4. Since the value of the radical is negative the alternative test must be used. The results are shown here: Predictor Coef SE Coef T P Constant –8.64 52.12 –0.17 0.869 X –0.0039 0.2004 –0.02 0.985 lagY 0.0497 0.2517 0.20 0.844 LagRes1 –0.0686 0.2957 –0.23 0.818
The lagged residual is not significant so no autocorrelation is suspected.
SUMS 52 337.1111111 2.432726593 243.2727 Moving average is probably not a suitable method because the data are trended upward. MAPE = 243.27/11 = 22.115. MAD = 52/11 = 4.72. MSD = 337.11/ 11 = 30.65.