# Regression result for second model still shows insignificant variables

I run a regression for my study. I used a quarterly data included 33 number of observations. my problem is independent variables are significant but not perfectly significant. when i run a multicollinearity, 2 of independent variables ,multicollinearity. so i exclude those two variables. and run a second model with only the significant variables. however, then my second model shows another independent variables as insignificant, while previous result it is shows a significant at 5% level. Sorry i don't know how to explain. hope you guys get what I am trying to solve. It really stress thinking the solutions. this is my result. I change the unit to % using formula 100*dlog(x). I am really thankful for any reply.

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I would certainly have expected that we already had a question on "why do $p$ values change if I change my model", but apparently we don't. So, here goes:

Remember how $p$ values and significance are calculated in regression:

1. You estimate coefficients
2. You estimate the standard errors of these coefficients
3. You divide the absolute values of the coefficients by the corresponding standard error to obtain $t$ statistics
4. You look up the $p$ value corresponding to your $t$ statistics

Now, if you change your model - by adding, deleting or transforming predictors - the first two calculations will change, because both a regressor's coefficient estimate and its standard error estimate are calculated in the context of the whole model.

And since the first two calculations change, which are the inputs to the last two steps, the final $p$ value will also change.

Different model $\longrightarrow$ different estimates $\longrightarrow$ different $p$ values.

And there is no reason why a $p<0.05$ for a given coefficient estimate before your model change should still be $<0.05$ after your model change.

So: everything works as it should. You will still need to interpret your results in the light of your theory.

Incidentally, what you are doing is called (I have added the tag). It is not good practice in inferential statistics, since $p$ values calculated after stepwise regression are invalid.

• One cannot stress the last point often enough. Throwing out variables or adding them on based on whether p-values are significant (or whether it improves AIC, AICC, BIC or some other criteria) and then fitting the model as if one had not done model selection (this qualifier is crucial - there are approaches that try to account for it) is a really bad approach. – Björn May 23 '16 at 6:52
• hence, does my regression is not valid? should I use log and not proceed to second model. or this has no solution? I am pleased if you could elaborate more on stepwise regression. Thank you for the replies – adah May 23 '16 at 7:11
• To interpret $p$ values, the model needs to be specified before you observe any data. However, if you are not interested in inference, but only in prediction, stepwise regression can be useful. – Stephan Kolassa May 23 '16 at 9:15