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Economics 310 Lecture 2 General Linear Model Continued.

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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. We want to choose an estimate such that the sum of squared differences between each observation y t and the estimate of the mean function E[y t ] is a minimum.

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LS Objective function

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First Order Conditions

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First Order Conditions Cont.

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First Order Conditions in matrix form

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Least-squares solution

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General Solution

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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

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Results of Matrix Example VARIANCE OF THE ESTIMATE-SIGMA**2 = STANDARD ERROR OF THE ESTIMATE-SIGMA = VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 16 DF P-VALUE CORR. COEFFICIENT AT MEANS X X X CONSTANT |_* generate results using matrix algebra |_genr x0=1 |_copy x0 x1 x2 x3 x |_matrix b=inv(x'x)*x'*y |_print b B

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Percapita Income Example sample 1 20 format (A11,F7.0,F8.0,2F6.0) read COUNTRY PCINC AGR IND SER /format CANADA SWEEDEN SWITZERLAND LUXEMBOURG U. KINGDOM DENMARK W. GERMANY FRANCE BELGUIM NORWAY ICELAND NETHERLANDS AUSTRIA IRELAND ITALY JAPAN GREECE SPAIN PORTUGAL TURKEY ols pcinc agr ind end

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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 = R-SQUARE ADJUSTED = VARIANCE OF THE ESTIMATE-SIGMA**2 = STANDARD ERROR OF THE ESTIMATE-SIGMA = SUM OF SQUARED ERRORS-SSE= E+07 MEAN OF DEPENDENT VARIABLE = LOG OF THE LIKELIHOOD FUNCTION = VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 17 DF P-VALUE CORR. COEFFICIENT AT MEANS AGR IND CONSTANT |_end

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Estimating the error variance

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Heuristic Explanation of unbiasness Truth Estimate Y X

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Sampling Properties

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Variance – Covariance Matrix

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