The Simple Linear Regression Model

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

2 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 Error C Total Root MSE R-square Dep Mean Adj R-sq C.V Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP X SSE K-1 SSR T-K T-1 SST R2 adjusted R2

3 Explaining Variation in yt
Açıklanamayan Toplam Sapma Açıklanabilen

4 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 ^

5 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

6 (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 =

7 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 = ^

8 (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 = ^

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

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

11 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

12 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

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

14 Regression Computer Output
Typical computer output of regression estimates: Table Computer Generated Least Squares Results (1) (2) (3) (4) (5) Parameter Standard t-stat Variable Estimate Error p-value INTERCEPT X

15 Regression Computer Output
b1 = b2 = se(b1) = var(b1) = = ^ se(b2) = var(b2) = = ^ se(b1) t = = = 1.84 b1 se(b2) b2 t = = = 0.1283 0.0305

16 Regression Computer Output
Sources of variation in the dependent variable: Table Analysis of Variance Table Sum of Mean Source DF Squares Square AKT=Explained KKT=Unexplained BKT=Total R-square:

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

18 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.

19 Reporting Regression Results
yt = xt (s.e.) ( ) (0.0305) yt = xt (t) (1.84) (4.20)

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.

21 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.

22 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

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

24 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

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

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

27 Reciprocal Models y x

28 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

29 Log-Log Models y x

30 Log-Log Models y x

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

32 Log-Linear Models y x

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

34 y Linear-Log Models x

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

36 Log-Inverse Model y x

37 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)

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

39 * 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

40 * 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

41 * 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

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

43 * 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