Temple Universtity

Department of Economics

Homework 4: Workouts in OLS

- It is pretty straightforward to show that the alternative estimator is unbiased.
- A. You can show this several different ways:

By expanding the multiplication and taking expectations it is obvious that is unbiased.

Now show that has a larger variance.

Numerically this is

The variance for the OLS estimator is

Although both estimators are unbiased, OLS has the smaller variance in the positive definite sense. This result is not surprising, given our knowledge of the Gauss-Markov Theorem.

- Compute the inner product of x
_{1}and x_{2}to show that they are not orthogonal - Compute the coefficients in the simple regression of x
_{2}on x_{1}and show that the residual sum of squares is not zero. LS essentially tries to find the linear relationship between two variables. - Compute the determinant of the augmented matrix made from the column vectors x
_{1}and x_{2}.

Orthogonality is sufficient but not necessary for linear independence.

We already knew that the off-diagonals wouldn't be zero, but it is still possible for the determinant to be zero, which would indicate linear dependence in the columns of x.

2.b. The OLS estimates, using the original data are

2.c. After adding 5 to each obs on y and 10 to each obs on x the OLS estimates are now

Notice that the intercept changed, but neither of the slopes is changed.

2.d. After multiplying each observation on the dependent variable by 10 the OLS coefficients change to:

2.e. If x_{1} is multiplied by 10 and the data is otherwise unchanged then the
OLS results change to:

2.f. The effect of adding a constant to both sides of the equation is to change the
intercept. The effect of multiplying the dependent variable by 10 is to rescale all of the
coefficients by a power of 10. Multiplying just one of the independent variables by 10 has
the effect of rescaling only the corresponding coefficient. Notice also, in your
regression output, that R^{2}, t-statistics and the F-statistic are
unaffected by rescaling the data.

3.b. The estimated coefficients are

Ordinary least squares regression. Dep. Variable = LNRENT Observations = 82 Mean of LHS = 0.2020316E+01 StdDev of residuals= 0.3841586E+00 R-squared = 0.9084267E+00 F[ 8, 73] = 0.9052197E+02 Log-likelihood = -0.3313692E+02 Amemiya Pr. Criter.= 0.1027730E+01 Durbin-Watson stat.= 2.2420910 |
Ordinary least squares regression. Weights = ONE Std.Dev of LHS = .1205161E+01 Sum of squares = 0.1077318E+02 Adjusted R-squared= .89839E+00 Restr.(á=0) Log-l = -0.13115E+03 Akaike Info.Crit. = 0.1637754E+00 Autocorrelation = -0.1210455 |
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ANOVA Source Regression Residual Total |
Variation 0.1068723E+03 0.1077318E+02 0.1176455E+03 |
D of Fr 8 73 81 |
Mean Square 0.1335903E+02 0.1475778E+00 0.1452413E+01 |
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Variable | Coefficients | Std Error | t-ratio | Pr|t|>x | Mean of X | Std Dev of x | |

Constant D61 D62 D63 D64 D65 LNMULT LNMEM LNACCES |
-0.10446 -0.13980 -0.48911 -0.59385 -0.92482 -1.1632 -.6536E-01 0.57933 -0.14060 |
0.3149 0.1665 0.1738 0.1661 0.1663 0.1661 .284E-01 .353E-01 .293E-01 |
-0.332 -0.840 -2.815 -3.575 -5.561 -7.003 -2.301 16.369 -4.794 |
0.7410 0.4037 0.0062 0.0006 0.0000 0.0000 0.0242 0.0000 0.0000 |
0.14634 0.13415 0.18293 0.21951 0.19512 4.7598 5.7263 1.8266 |
0.35562 0.34291 0.38899 0.41646 0.39873 2.6499 1.5140 2.3422 |

3.b. The quality adjusted price index series is:

The quality adjusted price index is declining throughout.

3.c. The results for the new model are:

Ordinary least squares regression. Dep. Variable = LNRENT Observations = 82 Mean of LHS = 0.2020316E+01 StdDev of residuals= 0.4907885E+00 R-squared = 0.8423462E+00 F[ 4, 77] = 0.1028530E+03 Log-likelihood = -0.5541066E+02 Amemiya Pr. Criter.= 0.1473431E+01 Durbin-Watson stat.= 1.5506978 |
Ordinary least squares regression. Weights = ONE Std.Dev of LHS = .1205161E+01 Sum of squares = 0.1854725E+02 Adjusted R-squared= 0.8341564E+00 Restr.(á=0) Log-l = -0.1311522E+03 Akaike Info.Crit. = 0.2555608E+00 Autocorrelation = 0.2246511 |
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ANOVA Source Regression Residual Total |
Variation 0.9909821E+02 0.1854725E+02 0.1176455E+03 |
Deg of Freedom 4 77 81 |
Mean Square 0.2477455E+02 0.2408734E+00 0.1452413E+01 |
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Variable | Coefficients | Std Error | t-ratio | Pr|t|>x | Mean of X | Std Dev of x | |

Constant - DUMM1 LNMULT LNMEM LNACCES |
-0.57377 -0.60048 -.3732E-01 0.61148 -0.11087 |
0.3856 0.1762 0.353E-01 0.439E-01 0.369E-01 |
-1.488 -3.407 -1.055 13.926 -2.999 |
0.1408 .00105 0.2947 0.0000 0.0036 |
0.8780 4.7598 5.7263 1.8266 |
0.32924 2.6499 1.5140 2.3422 |

The result of using a single dummy for the period 61 to 65 is to increase the magnitude
of each of the slope coefficients, to reduce both the R^{2} and the adjusted R^{2
}, and the coefficient on LNMULT is no longer different from zero.