Statistical inference -- Panden's econometric notes

Inspection of single overall parameters

t test

Inspection steps

1. The null hypothesis $H_0:{\beta }_j=0$
2. Determine significance level $\alpha$
3. Calculation t statistic
   $$t\equiv \frac{\widehat{{\beta }_j}-0}{se(\widehat{{\beta }_j})}$$
   Among them ${\beta }_j=\widehat{{\rho }_{xy}}\frac{\widehat{{\sigma }_x}}{\widehat{{\sigma }_y}}(Oneelementloveshape),\beta =(X^{T}X{)}^{-1}{X}^{T}Y(manyelementloveshape);se(\widehat{{\beta }_j})=\frac{\hat{\sigma }}{\sqrt{SS{T}_x}}(Oneelementloveshape),\hat{\sigma }=\frac{SSR}{n-2}$
4. Determine the critical value $t_{\frac{\alpha }{2}}(n-k-1)doubleSide,t_{\alpha }(n-k-1)singleSide$
5. Make an inference , If t If the statistic is greater than the critical value, the original hypothesis is rejected , Otherwise, I won't refuse

p Value method

In addition to comparing with the critical value , You can also directly calculate t Statistical p value , For both sides
$$p=P(|t_j|>|t_{\frac{\alpha }{2}}(n-k-1)|)$$

P The smaller the value, the more rejected ,P Once the value is less than the significance level, reject the original hypothesis

confidence interval approach

Inspection steps

1. The null hypothesis $H_0:{\beta }_j=0$
2. Determine significance level $\alpha$
3. utilize $\frac{\widehat{{\beta }_j}-{\beta }_j}{se(\widehat{{\beta }_j})}$ Obey the degree of freedom as n-k-1 Of t The fact of distribution , Construct confidence intervals
   $$[\widehat{{\beta }_j}-t_{\frac{\alpha }{2}}(n-k-1)se(\widehat{{\beta }_j}),\widehat{{\beta }_j}+t_{\frac{\alpha }{2}}(n-k-1)se(\widehat{{\beta }_j})]$$
4. Make statistical inferences , If the confidence interval is trapped 0, Then don't reject the original hypothesis , Otherwise, reject the original hypothesis

Be careful For one 95% confidence interval , If you don't reject the original hypothesis , Can you say that he is 95% The probability of contains the truth ？

You can't , The confidence interval either contains a true value or does not contain a true value ,95% It's just that 100 Next time , Yes 95 Times contain truth values

Test of multiple linear constraints

F test

Inspection steps

1. The null hypothesis $H_0:{\beta }_3=0,{\beta }_4=0,{\beta }_5=0,\dots$
2. Determine significance level $\alpha$
3. structure F statistic , Constrained models are used respectively ( Removed $x_3,x_4,x_5,\dots$) And unconstrained models ( The original model )
   $$F\equiv \frac{SS{R}_r-SS{R}_{ur}}{SS{R}_{ur}}\cdot \frac{n-k-1}{q}$$
   among q by $x_3,x_4,x_5,\dots$ The number of
4. Determine the critical value $F_{\alpha }(q,n-k-1)(onlyYessingleSide)$
5. Make statistical inferences , Reject the original hypothesis once it is greater than the critical value , Otherwise, I won't refuse

F Tested $R^2$

F Statistics can also be written in the following form
$$F=\frac{R_{ur}^2-R_r^2}{1-R_{ur}^2}\cdot \frac{n-k-1}{q}$$
among $R_{ur}^2$ It's from the original model $R^2$

adjustment $R^2$

Besides using F Test to select nested models , You can also use to adjust $R^2$ For non nested models ( Of course, nested models can also ) Make a selection
$${\bar{R}}^2=1-\frac{SSR}{SST}\cdot \frac{n-1}{n-k-1}$$
It can also be based on $R^2$ To calculate
$${\bar{R}}^2=1-(1-R^2)\cdot \frac{n-1}{n-k-1}$$

p Value method

F Inspection can also be done with P Value method , Consistent with the above operation

The regression is significant as a whole

Specially , When all parameters are included in the original assumption ${\beta }_1,\cdots ,{\beta }_k$ when , That is to test whether all explanatory variables explain the explained variables ( perhaps $R^2$ Significantly different from 0 When )
$$F=\frac{R^2}{1-R^2}\cdot \frac{n-k-1}{k}$$

Large sample test

It should be noted that : Ahead t Test and F Inspection is to meet MLR.1-5 Hypothetical , One of the important assumptions is the same variance , Once the same variance is not satisfied, it is no longer applicable ; But it still applies in large samples , In addition, there are Lagrange multipliers (LM) statistic

Joint test LM statistic

Inspection steps

1. The null hypothesis $H_0:{\beta }_3=0,{\beta }_4=0,{\beta }_5=0,\dots$
2. take y The independent variables after excluding constraints are regressed , And save the residuals $\tilde{u}$
3. take $\tilde{u}$ All independent variables were regressed , obtain $R^2$, Write it down as $R_u^2$
4. Calculation LM statistic
   $$LM=n{R}_u^2$$
5. According to the significance level $\alpha$, Determine the critical value ${\chi }_{\alpha }^2(q)$
6. LM If the statistic is greater than the critical value, reject , Otherwise, I won't refuse