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Economics 310

Economics 310 . Lecture 2 General Linear Model Continued. Estimation of Linear Model. To find an estimator of the unknown coefficients in the vector  , we follow the least squares procedure. A given estimate of  gives an estimate of the mean function E[y t ]= 1 +x t2  2 +x t3  3 .

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Economics 310

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  1. Economics 310 Lecture 2 General Linear Model Continued.

  2. Estimation of Linear Model • To find an estimator of the unknown coefficients in the vector , we follow the least squares procedure. • A given estimate of  gives an estimate of the mean function E[yt]=1+xt22+xt33. • We want to choose an estimate such that the sum of squared differences between each observation yt and the estimate of the mean function E[yt] is a minimum.

  3. LS Objective function

  4. First Order Conditions

  5. First Order Conditions Cont.

  6. First Order Conditions in matrix form

  7. Least-squares solution

  8. General Solution

  9. OLS – Shazam vs Matrix Alegbra sample 1 20 read x1 x2 x3 * generate a set of y for y=10+3x1-4x2+5x3+e genr y=10+3*x1-4*x2+5*x3+nor(2) ols y x1 x2 x3 * generate results using matrix algebra genr x0=1 copy x0 x1 x2 x3 x matrix b=inv(x'x)*x'*y print b stop

  10. Results of Matrix Example VARIANCE OF THE ESTIMATE-SIGMA**2 = 135.55 STANDARD ERROR OF THE ESTIMATE-SIGMA = 11.643 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 16 DF P-VALUE CORR. COEFFICIENT AT MEANS X1 2.9701 0.1076 27.61 0.000 0.990 0.3840 0.7246 X2 -4.0604 0.1318 -30.82 0.000-0.992 -0.3787 -1.0227 X3 5.0685 0.1244 40.75 0.000 0.995 0.5907 1.2387 CONSTANT 11.487 9.309 1.234 0.235 0.295 0.0000 0.0594 |_* generate results using matrix algebra |_genr x0=1 |_copy x0 x1 x2 x3 x |_matrix b=inv(x'x)*x'*y |_print b B 11.48749 2.970102 -4.060408 5.068454

  11. Percapita Income Example sample 1 20 format (A11,F7.0,F8.0,2F6.0) read COUNTRY PCINC AGR IND SER /format CANADA 1536 13 43 45 SWEEDEN 1644 14 53 33 SWITZERLAND 1361 11 56 33 LUXEMBOURG 1242 15 51 34 U. KINGDOM 1105 4 56 40 DENMARK 1049 18 45 37 W. GERMANY 1035 15 60 25 FRANCE 1013 20 44 36 BELGUIM 1005 6 52 42 NORWAY 977 20 49 32 ICELAND 839 25 47 29 NETHERLANDS 810 11 49 40 AUSTRIA 681 23 47 30 IRELAND 529 36 30 34 ITALY 504 27 46 28 JAPAN 344 33 35 32 GREECE 324 56 24 20 SPAIN 290 42 37 21 PORTUGAL 238 44 33 23 TURKEY 177 79 12 9 ols pcinc agr ind end

  12. Shazam OLS Results |_sample 1 20 |_format(A13,F5.0,3F3.0) |_read COUNTRY PCINC AGR IND SER / format READ USES FORMAT:(A13,F5.0,3F3.0) |_ols pcinc agr ind REQUIRED MEMORY IS PAR= 2 CURRENT PAR= 2000 OLS ESTIMATION 20 OBSERVATIONS DEPENDENT VARIABLE= PCINC ...NOTE..SAMPLE RANGE SET TO: 1, 20 R-SQUARE = 0.6231 R-SQUARE ADJUSTED = 0.5788 VARIANCE OF THE ESTIMATE-SIGMA**2 = 67294. STANDARD ERROR OF THE ESTIMATE-SIGMA = 259.41 SUM OF SQUARED ERRORS-SSE= 0.11440E+07 MEAN OF DEPENDENT VARIABLE = 767.10 LOG OF THE LIKELIHOOD FUNCTION = -137.922 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 17 DF P-VALUE CORR. COEFFICIENT AT MEANS AGR -14.205 8.219 -1.728 0.102-0.387 -0.6912 -0.5352 IND 3.5139 13.42 0.2618 0.797 0.063 0.1047 0.1926 CONSTANT 1029.9 792.0 1.300 0.211 0.301 0.0000 1.3425 |_end

  13. Estimating the error variance

  14. Heuristic Explanation of unbiasness Y Estimate Truth X

  15. Sampling Properties

  16. Variance – Covariance Matrix

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