Assume Y = X β 0 + ϵ where ϵ is zero mean and X is fixed. I know that under certain conditions on the design matrix X in OLS, the sample mean of the residuals e ¯ is 0. Can we say the same for the true population mean of residuals as well? Y ^ = X β ^ and E [ e ] = E ( Y − Y ^ ) = E [ X β 0 − X β ^ ] = X ( β 0 − E ( β ^ ) ) = 0since β ^ is an unbiased estimator of β 0 .Am I making a mistake?

Question

Assume   Y  =  X      β    0    +  ϵ where   ϵ is zero mean and   X is fixed. I know that under certain conditions on the design matrix   X in OLS, the sample mean of the residuals       e    ¯   is   0. Can we say the same for the true population mean of residuals as well?      Y    ^    =  X      β    ^   and  E  [  e  ]  =  E  (  Y  −      Y    ^    )  =  E  [  X      β    0    −  X      β    ^    ]  =  X  (      β    0    −  E  (      β    ^    )  )  =  0since       β    ^   is an unbiased estimator of       β    0  .Am I making a mistake?

slapadabassyc · Accepted Answer

If we are treating   X as fixed, which I take to mean non-random, then by assumption  E  [  ϵ  ]  =  0You are sort of going in a circle because you say   E  [  Y  ]  =  X      β    0  , but this is because of our prior assumption. It is a result not an assumption. We have                    Y                            =        X                  β          0                +        ϵ                            E        [        Y        ]                            =        E        [        X                  β          0                +        ϵ        ]                            E        [        Y        ]                            =        E        [        X                  β          0                ]        +        E        [        ϵ        ]                            E        [        Y        ]                            =        X                  β          0                +        0                            E        [        Y        ]                            =        X                  β          0                    where the fourth line is a result of our assumption and   X being non-random (as well as       β    0   being a constant).

Assume Y=X beta_0+epsilon where ϵ is zero mean and X is fixed. I know that under certain conditions on the design matrix X in OLS, the sample mean of the residuals e is 0. Can we say the same for the true population mean of residuals as well?

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