Get inspired by the success stories of our students in IIT JAM MS, ISI  MStat, CMI MSc DS.  Learn More 

Restricted Regression Problem | ISI MStat 2017 PSB Problem 7

Join Trial or Access Free Resources

This problem is a regression problem, where we use the ordinary least square methods, to estimate the parameters in a restricted case scenario. This is ISI MStat 2017 PSB Problem 7.


Consider independent observations \({\left(y_{i}, x_{1 i}, x_{2 i}\right): 1 \leq i \leq n}\) from the regression model
y_{i}=\beta_{1} x_{1 i}+\beta_{2} x_{2 i}+\epsilon_{i}, i=1, \ldots, n
$$ where \(x_{1 i}\) and \(x_{2 i}\) are scalar covariates, \(\beta_{1}\) and \(\beta_{2}\) are unknown scalar
coefficients, and \(\epsilon_{i}\) are uncorrelated errors with mean 0 and variance \(\sigma^{2}>0\). Instead of using the correct model, we obtain an estimate \(\hat{\beta_{1}}\) of \(\beta_{1}\) by minimizing
\sum_{i=1}^{n}\left(y_{i}-\beta_{1} x_{1 i}\right)^{2}
$$ Find the bias and mean squared error of \(\hat{\beta}_{1}\).



It is sort of a restricted regression problem because maybe we have tested the fact that \(\beta_2 = 0\). Hence, we are interested in the estimate of \(\beta_1\) given \(\beta_2 = 0\). This is essentially the statistical significance of this problem, and we will see how it turns out in the estimate of \(\beta_1\).

Let's start with some notational nomenclature.
\( \sum_{i=1}^{n} a_{i} b_{i} = s_{a,b} \)

Let's minimize \( L(\beta_1) = \sum_{i=1}^{n}\left(y_{i}-\beta_{1} x_{1 i}\right)^{2}\) by differentiating w.r.t \(\beta_1\) and equating to 0.

\( \frac{dL(\beta_1)}{d\beta_1}\sum_{i=1}^{n}\left(y_{i}-\beta_{1} x_{1 i}\right)^{2} = 0\)

\( \Rightarrow \sum_{i=1}^{n} x_{1 i} \left(y_{i}-\beta_{1} x_{1 i}\right) = 0 \)

\( \Rightarrow \hat{\beta_1} = \frac{s_{x_{1},y}}{s_{x_{1},x_{1}}} \)

From, the given conditions, \( E(Y_{i})=\beta_{1} X_{1 i}+\beta_{2} X_{2 i}\).

\( \Rightarrow E(s_{X_{1},Y}) = \beta_{1}s_{X_{1},X_{1}} +\beta_{2} s_{X_{1},X_{2}} \).

Since, \(x's\) are constant, \( E(\hat{\beta_1}) = \beta_{1} +\beta_{2} \frac{s_{X_{1},X_{2}}}{s_{X_{1},X_{1}}} \).

\( Bias(\hat{\beta_1}) = \beta_{2} \frac{s_{X_{1},X_{2}}}{s_{X_{1},X_{1}}} \).

Thus, observe that the more \( \beta_2 \) is close to 0, the more bias is close to 0.

From, the given conditions,

\( Y_{i} - \beta_{1} X_{1 i} - \beta_{2} X_{2 i}\) ~ Something\(( 0 , \sigma^2\)).

\( \hat{\beta_1} = \frac{s_{x_{1},y}}{s_{x_{1},x_{1}}}\) ~ Something\(( E(\hat{\beta_{1}}) , Var(\hat{\beta_1}))\).

\(Var(\hat{\beta_1}) = \frac{\sum_{i=1}^{n} x_{1i}^2 Var(Y_{i})}{s_{X_1, X_1}^2} = \frac{\sigma^2}{s_{X_1, X_1}} \)

\( MSE(\hat{\beta_1}) = Variance + \text{Bias}^2 = \frac{\sigma^2}{s_{X_1, X_1}} + \beta_{2}^2(\frac{s_{X_{1},X_{2}}}{s_{X_{1},X_{1}}})^2\)

Observe, that even the MSE is minimized if \(\beta_2 = 0\).

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
ISI Entrance Solutions
ISI CMI Self Paced