MULTIPLE REGRESSION ANALYSIS THE THREE VARIABLE MODEL: NOTATION AND ASSUMPTION 08/06/2015Ika Barokah S

Multiple Regression Analysis Dalam banyak hal, analisis regresi sederhana tidak bisa (cukup) menjelaskan variasi Y secara akurat  R 2 yang relatif rendah  perlu penambahan variabel explanatory pada model (fungsi) Model standar : Y i = b 1 + b 2 X 2i + b 3 X 3i + … +b k X ki + e i i = 1,2,3…n k = banyaknya x, k =2,3,…m dimana : b 2 = perubahan nilai Y per satu satuan perubahan X 2i dengan asumsi X 3i konstan b 3 = perubahan nilai Y per satu satuan perubahan X 3i dengan asumsi X 2i konstan 08/06/2015 Ika Barokah S

NOTATION Y i =  1 +  2 X 2i +  3 X 3i + U i ASSUMPTIONS Zero mean value of U i Zero mean value of U i No serial correlation No serial correlation Homoscedasticity Homoscedasticity Zero covariance between U i and X i Zero covariance between U i and X i No specification bias No specification bias No exact collinearity between the X variables No exact collinearity between the X variables 08/06/2015 Ika Barokah S

.. x1x1 x2x2 The homoscedastic normal distribution with a single explanatory variable E(y|x) =  0 +  1 x y f(y|x) Normaldistributions 08/06/2015 Ika Barokah S

ESTIMATION 08/06/2015 Ika Barokah S

Beta Coefficients Occasional you’ll see reference to a “standardized coefficient” or “beta coefficient” which has a specific meaning Occasional you’ll see reference to a “standardized coefficient” or “beta coefficient” which has a specific meaning Idea is to replace y and each x variable with a standardized version – i.e. subtract mean and divide by standard deviation Idea is to replace y and each x variable with a standardized version – i.e. subtract mean and divide by standard deviation Coefficient reflects standard deviation of y for a one standard deviation change in x Coefficient reflects standard deviation of y for a one standard deviation change in x 08/06/2015 Ika Barokah S

VARIANCE AND STANDARD ERROR 08/06/2015 Ika Barokah S

Koefisien Determinasi (R 2 ) Mengukur ketepatan /kecocokan (goodness of fit) Mengukur proporsi/presentase sumbangan x terhadap variasi y Mengukur besarnya proporsi/presentase sumbangan x 2 da x 3 terhadap variasi y secara bersama-sama (R 2 ) R 2 = ESS = ∑Ŷ 2 TSS ∑Y 2 = b 2 ∑X 2i. Y+ b 3 ∑X 3i.Y ∑Y 2 08/06/2015 Ika Barokah S

Sifat-sifat R 2 Nilai R 2  0 ≤ R 2 ≤ 1 R 2  non decreasing function (fungsi yang mempunyai nilai positif) Semakin banyak/ setiap penambahan veriabel bebas (x) kedalam model regresi  R 2 juga meningkat 08/06/2015 Ika Barokah S

Adjusted R 2 Adjusted R 2 adalah R 2 yang sudah disesuaikan dengan df dari masing- masing jumlah kuadrat yang tercakup dalam perhitungan adjusted R 2 Adjusted R 2 = 1-(1- R 2 ) (n-1) (n-k) Beberapa hal tentang Adjusted R 2 :  Jika k>1, Adjusted R 2 < R 2  Adjusted R 2 dapat bernilai negatif, meskipun R 2 non negatif 08/06/2015 Ika Barokah S

The t Test 08/06/2015 Ika Barokah S

Uji Hipotesa digunakan untuk menguji statement tertentu tentang populasi Langkah-langkah dalam uji t : i. Memformulasikan H o dan H a H o : b i = 0 H a : b i ≠ 0 ii. Menghitung distribusi probabilitas : t hitung = b i Sb i iii. Memilih level of significant α  1%; 5%; 10%  t tabel t α/2, n-k iv. Keputusan : Terima H o : ii < iii Tolak H o : ii > iii 08/06/2015 Ika Barokah S

y i =  0 +  1 x i1 + … +  k x ik + u i H 0 :  j = 0 H 1 :  j > 0 c 0   One-Sided Alternatives Fail to reject reject 08/06/2015 Ika Barokah S

y i =  0 +  1 X i1 + … +  k X ik + u i H 0 :  j = 0 H 1 :  j ≠ 0 c 0   -c  Two-Sided Alternatives reject fail to reject 08/06/2015 Ika Barokah S

Verbally, R-square measure the proportion or percentage of the total variation in Y explained by the regression model. THE COEFFICIENT OF DETERMINATION A MEASURE OF “GOODNESS OF FIT” 08/06/2015 Ika Barokah S

Adjusted R-Squared Recall that the R 2 will always increase as more variables are added to the model The adjusted R 2 takes into account the number of variables in a model, and may decrease 08/06/2015 Ika Barokah S

It’s easy to see that the adjusted R 2 is just (1 – R 2 )( n – 1) / ( n – k – 1), but most packages will give you both R 2 and adj- R 2 You can compare the fit of 2 models (with the same y ) by comparing the adj- R 2 You cannot use the adj- R 2 to compare models with different y ’s (e.g. y vs. ln( y )) 08/06/2015 Ika Barokah S

Langkah-langkah dalam uji F : i. Memformulasikan H o dan H a H o : b 1 =b 2 =b 3 =….b k = 0 H a : setidaknya salah satu b i ≠ 0 ii. Menghitung distribusi probabilitas : F hitung = RSS/(k-1) ESS/(n-k) iii. Memilih level of significant α  1%;5%;10%  F tabel F tabel  F k-1;n-k iv. Keputusan : Terima H o : ii < iii Tolak H o : ii > iii 08/06/2015 Ika Barokah S

0 c   f( F ) F The F statistic reject fail to reject Reject H 0 at  significance level if F > c 08/06/2015 Ika Barokah S

08/06/2015 Ika Barokah S