The Simple Linear Regression Model Bölüm 3 The Simple Linear Regression Model

Regression Output--A SAS SSR/(K-1) Model: MODEL1 Dependent Variable: Y Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 1 25221.22299 25221.22299 17.647 0.0002 Error 38 54311.33145 1429.24556 C Total 39 79532.55444 Root MSE 37.80536 R-square 0.3171 Dep Mean 130.31300 Adj R-sq 0.2991 C.V. 29.01120 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 40.767556 22.13865442 1.841 0.0734 X 1 0.128289 0.03053925 4.201 0.0002 SSE K-1 SSR T-K T-1 SST R2 adjusted R2

Explaining Variation in yt Açıklanamayan Toplam Sapma Açıklanabilen

yt = b1 + b2xt + et yt = b1 + b2xt et = yt  yt = yt  b1 b2xt ^ ^ ^ Explaining Variation in yt degerine ait sapmanın açıklanması yt = b1 + b2xt + et ^ yt = b1 + b2xt ^ Açıklanan Degişken Açıklanamayan Sapma et = yt  yt = yt  b1 b2xt ^

Explaining Variation in yt ^ ^ yt = yt + et using y as baseline yt  y = yt  y + et ^ Karesini alıp toplarsak cross product term drops out (yty)2 = (yty)2 +et t = 1 T ^ 2 SST = SSR + SSE BKT= AKT+KKT

(yt y)2 SST = Total Variation in yt SST = total sum of squares BKT=bütün kareler toplamı SST,yt degerinin sapmasını yani y Etrafında dagılımı ölçer (yt y)2 t = 1 T SST =

Explained Variation in yt SSR = regression sum of squares AKT=Açıklanan Kareler Toplamı yt = b1 + b2xt ^ Fitted yt values: ^ SSR tahmin edilen yt nin y Etrafındaki dagılımını ölçer ^ (yt y)2 t = 1 T SSR = ^

(Yt) Açıklanamayan Sapmaya dair SSE = error sum of squares (Açıklanamayan hata) KKT=Açıklanamayan Kalıntı Kareler Toplamı ^ ^ et = ytyt = yt b1  b2xt SSE gerçek degerler ile yt Tahmin yt degerleri Arasındaki farklılıgı ölçer ^ (yt yt)2 = et2 t = 1 T SSE = ^

Analysis of Variance Table Table 6.1 Analysis of Variance Table Source of Sum of Mean Variation DF Squares Square Explained 1 SSR SSR/1 Unexplained T-2 SSE SSE/(T-2) [= 2] Total T-1 SST ^

R2 = 0 < R2 < 1 SSR SST Coefficient of Determination yt degerinden sapmanın standard ölçümü: 0 < R2 < 1 SSR SST R2 =

R2 = = 1  Coefficient of Determination = + 1 = + SSR SSE SST SST = SSR + SSE SST SSR SSE SST SST SST = + SST ile bölersek SSR SSE SST SST 1 = + SSR SST R2 = = 1  SSE

Coefficient of Determination R2 sadece descriptive (tanımlayıcı Ölçüm yapar Kalite ölçüm birimi degildir measure.Yani regresyonun Ölçüm kalitesini göstermez maximizing R2 yani en yüksek Deger tek başına buna bakmak dogru degıldir

Regression Output--B Excel SSR=AKT T SSE=KKT varyansı SST=BKT b2 Se(b2) Interval Estimate For hypothesis testing

Regression Computer Output Typical computer output of regression estimates: Table 6.2 Computer Generated Least Squares Results (1) (2) (3) (4) (5) Parameter Standard t-stat Variable Estimate Error p-value INTERCEPT 40.7676 22.1387 1.841 0.0734 X 0.1283 0.0305 4.201 0.0002

Regression Computer Output b1 = 40.7676 b2 = 0.1283 se(b1) = var(b1) = 490.12 = 22.1287 ^ se(b2) = var(b2) = 0.0009326 = 0.0305 ^ se(b1) t = = = 1.84 b1 40.7676 22.1287 se(b2) b2 t = = = 4.20 0.1283 0.0305

Regression Computer Output Sources of variation in the dependent variable: Table 6.3 Analysis of Variance Table Sum of Mean Source DF Squares Square AKT=Explained 1 25221.2229 25221.2229 KKT=Unexplained 38 54311.3314 1429.2455 BKT=Total 39 79532.5544 R-square: 0.3171

Regression Computer Output SST = (yty)2 = 79532 SSR = (yty)2 = 25221 ^ SSE = et2 = 54311 ^ SSE /(T-2) = 2 = 1429.2455 ^ SSR SST R2 = = 1 = 0.317 SSE

R2 = = 1 = 0.317 Yorumu? SSR SSE SST Y degerinin ortalamasının etrafındaki farklılık regresyon modeli ile yüzde 31.7% oranında açıklana bilmiştir Yada bir diger ifade ile modelimiz Y degeri üzerinde yüzde 31.7 oranında açıklayıcı gücü vardır.

Reporting Regression Results yt = 40.7676 + 0.1283xt (s.e.) (22.1387) (0.0305) yt = 40.7676 + 0.1283xt (t) (1.84) (4.20)

Reporting Regression Results This R2 value may seem low but it is typical in studies involving cross-sectional data analyzed at the individual or micro level. A considerably higher R2 value would be expected in studies involving time-series data analyzed at an aggregate or macro level.

Functional Forms The term linear in a simple regression model does not mean a linear relationship between variables, but a model in which the parameters enter the model in a linear way.

yt = 1 + 2xt + exp(3xt) + et Linear vs. Nonlinear Linear Statistical Models: yt = 1 + 2xt + et yt = 1 + 2 ln(xt) + et ln(yt) = 1 + 2xt + et yt = 1 + 2xt + et 2 Nonlinear Statistical Models: yt = 1 + 2xt + et 3 yt = 1 + 2xt + et 3 yt = 1 + 2xt + exp(3xt) + et

nonlinear relationship between food expenditure and income Linear vs. Nonlinear y nonlinear relationship between food expenditure and income food expenditure x income

1. Linear 2. Reciprocal 3. Log-Log 4. Log-Linear 5. Linear-Log Useful Functional Forms 1. Linear 2. Reciprocal 3. Log-Log 4. Log-Linear 5. Linear-Log 6. Log-Inverse Look at each form and its slope and elasticity

Linear yt = 1 + 2xt + et Useful Functional Forms xt slope: 2 elasticity: 2 yt

Reciprocal yt = 1 + 2 + et xt xt Useful Functional Forms 1 slope: elasticity: 1 xt 2 2 1 xt yt 2

Reciprocal Models y x

Log-Log (Constant Elasticity Model) Useful Functional Forms Log-Log (Constant Elasticity Model) ln(yt)= 1 + 2ln(xt) + et yt slope: 2 elasticity: 2 xt

Log-Log Models y x

Log-Log Models y x

slope: 2 yt elasticity: 2xt Useful Functional Forms Log-Linear ln(yt)= 1 + 2xt + et slope: 2 yt elasticity: 2xt

Log-Linear Models y x

Linear-Log _ yt= 1 + 2ln(xt) + et Useful Functional Forms 1 slope: 2 elasticity: 2 1 xt yt

y Linear-Log Models x

Log-Inverse ln(yt) = 1 - 2 + et xt Useful Functional Forms 1 slope: 2 elasticity: 2 x2t yt 1 xt

Log-Inverse Model y x

1. E (et) = 0 2. var (et) = 2 3. cov(ei, ej) = 0 4. et ~ N(0, 2) What Functional Form? 1. E (et) = 0 2. var (et) = 2 3. cov(ei, ej) = 0 4. et ~ N(0, 2)

1. Demand Models 2. Supply Models 3. Production Functions Economic Models 1. Demand Models 2. Supply Models 3. Production Functions 4. Cost Functions

* quality demanded (yd) and price (x) Economic Models Demand Models * quality demanded (yd) and price (x) * constant elasticity ln(yt )= 1 + 2ln(x)t + et d

* quality supplied (ys) and price (x) Economic Models Supply Models * quality supplied (ys) and price (x) * constant elasticity ln(yt )= 1 + 2ln(xt) + et s

* output (y) and input (x) Economic Models Production Functions * output (y) and input (x) * constant elasticity Cobb-Douglas Production Function: ln(yt)= 1 + 2ln(xt) + et

* total cost (y) and output (x) Economic Models Cost Functions * total cost (y) and output (x) yt = 1 + 2x2t + et

* average cost (x/y) and output (x) Economic Models Cost Functions * average cost (x/y) and output (x) (yt/xt) = 1/xt + 2xt + et/xt