L2.pdf - APPLIED STATISTICS Inferential Tools for Simple Linear Regression Dr Tao Zou Research School of Finance Actuarial Studies Statistics The

L2.pdf - APPLIED STATISTICS Inferential Tools for Simple...

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APPLIED STATISTICS Inferential Tools for Simple Linear Regression Dr Tao Zou Research School of Finance, Actuarial Studies & Statistics The Australian National University Last Updated: Wed Aug 9 16:17:16 2017 1 / 33
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Overview Sampling Distribution of Estimation Standard Error of Estimation Hypothesis Testing Confidence Intervals and Prediction Intervals 2 / 33
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References 1. F.L. Ramsey and D.W. Schafer (2012) Chapter 7 of The Statistical Sleuth 2. The slides are made by R Markdown . 3 / 33
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Distinguish Parameters and Estimation SLR model μ { Y | X } = β 0 + β 1 X and real data ( X 1 , Y 1 ) , · · · , ( X n , Y n ) . Parameters Estimation (notation hat " ˆ ") β 1 n i = 1 ( X i - ¯ X )( Y i - ¯ Y ) n i = 1 ( X i - ¯ X ) 2 (denoted by ˆ β 1 ) β 0 ¯ Y - ˆ β 1 ¯ X (denoted by ˆ β 0 ) unknown for real data can be computed based on real data the value is unique can be different for different datasets Hence, if we have another sample/dataset, e.g., ( X n + 1 , Y n + 1 ) , · · · , ( X n + n , Y n + n ) , we will obtain different realisations of ˆ β 0 and ˆ β 1 . 4 / 33
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Sampling Distributions of ˆ β 0 and ˆ β 1 The distributions of the realisations are the sampling distributions. We consider the sampling distributions of ˆ β 0 and ˆ β 1 given the values of the explanatory variables. It can be shown mathematically that the sampling distributions of ˆ β 0 and ˆ β 1 are both normal . 5 / 33
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Sampling Distributions of ˆ β 0 and ˆ β 1 (Con’d) Sampling distribution of ˆ β 1 : ˆ β 1 is normal distributed; Mean= E ( ˆ β 1 ) = β 1 ; Spread=SD( ˆ β 1 )= σ 1 ( n - 1 ) s 2 X . Sampling distribution of ˆ β 0 : ˆ β 0 is normal distributed; Mean= E ( ˆ β 0 ) = β 0 ; Spread=SD( ˆ β 0 )= σ 1 n + ¯ X 2 ( n - 1 ) s 2 X . Here, s 2 X = 1 n - 1 n i = 1 ( X i - ¯ X ) 2 . Knowing the sampling distributions allows us to make inferences about β 0 and β 1 . Remark : SLR model assumptions 1 & 2 & 3 can be described by Y = β 0 + β 1 X + E , where E ∼ N ( 0 , σ 2 ) . It follows Y N ( β 0 + β 1 X , σ 2 ) . 6 / 33
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Example: Simulation for SLR μ ( Y | X ) = β 0 + β 1 X . 1. Set the unknown parameters β 0 = 2 and β 1 = 1 by yourselves. rm ( list= ls ()) beta0= 2 ;beta1= 1 2. Randomly generate R = 1000 repeated samples of { Y i , X i } n i = 1 . For each sample, obtain the statistics required in your analysis. Here we consider the least squares estimates ˆ β 0 and ˆ β 1 as the statistics. n= 100 R= 1000 hatbeta0= rep ( 0 ,R) #space to store the different realisations of estimations hatbeta1= rep ( 0 ,R) set.seed ( 1 ) for(r in 1 :R) { X= 1 :n #our X values errors= rnorm (n) #generate a set of errors, which implies sigma=1 Y=beta0+beta1*X+errors #generate a set of response values SLRfit= lm (Y~X) #fit the SLR hatbeta0[r]=SLRfit$coef[ 1 ] #get the estimation hatbeta1[r]=SLRfit$coef[ 2 ] } 3. The sampling distribution of the statistics can be described by the R = 1000 different statistics values from R = 1000 repeated samples . 7 / 33
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“set.seed()” set.seed ( 1 ) head ( rnorm (n)) ## [1] -0.6264538 0.1836433 -0.8356286 1.5952808 0.3295078 -0.8204684 set.seed ( 2 ) head ( rnorm (n)) ## [1] -0.89691455 0.18484918 1.58784533 -1.13037567 -0.08025176 0.13242028 set.seed ( 1 ) head ( rnorm (n)) ## [1] -0.6264538 0.1836433 -0.8356286 1.5952808 0.3295078 -0.8204684 seed=1 seed=2 · · · -0.6264538 0.89691455 · · · 0.1836433 0.18484918 · · · -0.8356286 1.58784533 · · · 1.5952808 -1.13037567 · · · 0.3295078 -0.08025176 · · · -0.8204684 0.13242028 · · · .
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