# High R-squared although many insignificant coefficients

I just did a regression based on the gravity model where I try to identify the most important factors that determine the trade flows. In total I have 18 variables and 363 observations. In fact I would do several regressions not including all variables at once. Although I included them all just to have a look and I got an R-squared of 0,9162 (using robust standard errors). Due to poor data availability for certain variables I only have 102 observations by including all variables. The problem is that only 9 coefficients are significant at the 10% level. Can I now assume that only those variables have a significant impact on trade flows? Can I assume that the regression is good as R-squared is quite high? Or can I do some tests whether those results are reliable (so that I can use them in my research paper)?

I am doing this for the first time and I am a little bit confused.

There are a lot of things going on here, probably more than can be addressed in this forum. I will mention some of the major concerns, but you really should consult a regression textbook, class, or statistical consultant.

Due to missingness you are using less that a third of your observations. That should cause concern. Is the data missing at random? completely at random? or could the missingness be related to the variables in the model in an informative way? (if the last, then everything is suspect, if one of the former then you should really look at methods like multiple imputation or others to deal with missing values).

Each of the p-values is measuring the effect of that variable given that the other variables are in the model. It is possible that some of the "non-significant" variables are very important predictors, only that they are redundant given other (possibly also "non-significant") variables in the model. If you remove all the non-significant terms then it is possible that you would be removing some very important predictors.

It is not clear what question you are really trying to answer, and what to do next really depends on the question to be answered. If you are just trying to find a model for prediction, then something like "lasso" may be the next step (including cross-validation of your predictions). If you are interested if certain predictors are important given other predictors in the model, then a full-reduced test is appropriate. Other questions lead to other next steps, but to really understand it requires study and understanding, not just "this guy on the internet said ...".

There's some chance you're overfitting the data with this sample size (the $R^2$ value seems suspiciously high), but more to the point there's nothing inconsistent between a high $R^2$ and lots of "insignificant" predictors. This is simply because the coefficient of determination never decreases when you add variables to the model, so if you start with a high $R^2$ then you'll end up with one as long as you don't drop any variables. Below are some simple simulations that demonstrate this idea. Here I generated data according to the model $$Y_i = x_{1i} + \epsilon_i$$ where $\epsilon_i \sim$ normal$(0, \sigma^2)$ and then fit a regression only involving $x_{1i}$ along with another that included nine extra noise predictors $x_{2i}, x_{3i}, \ldots, x_{10i}$ that bore no relation to $Y_i$. We can see that the second model has a larger $R^2$ even though the added variables are not important.

Model 1:

                            OLS Regression Results
==============================================================================
Dep. Variable:                      y   R-squared:                       0.935
Method:                 Least Squares   F-statistic:                     1424.
Date:                Sun, 02 Aug 2015   Prob (F-statistic):           1.44e-60
Time:                        22:37:24   Log-Likelihood:                -4.2454
No. Observations:                 100   AIC:                             10.49
Df Residuals:                      99   BIC:                             13.10
Df Model:                           1
Covariance Type:            nonrobust
==============================================================================
coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
var_1          0.9865      0.026     37.734      0.000         0.935     1.038
==============================================================================
Omnibus:                        7.358   Durbin-Watson:                   1.957
Prob(Omnibus):                  0.025   Jarque-Bera (JB):                3.027
Skew:                           0.016   Prob(JB):                        0.220
Kurtosis:                       2.148   Cond. No.                         1.00
==============================================================================


Model 2:

                            OLS Regression Results
==============================================================================
Dep. Variable:                      y   R-squared:                       0.941
Method:                 Least Squares   F-statistic:                     143.2
Date:                Sun, 02 Aug 2015   Prob (F-statistic):           9.08e-51
Time:                        22:37:27   Log-Likelihood:                0.48280
No. Observations:                 100   AIC:                             19.03
Df Residuals:                      90   BIC:                             45.09
Df Model:                          10
Covariance Type:            nonrobust
==============================================================================
coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
var_1          0.9817      0.028     35.336      0.000         0.927     1.037
var_2         -0.0011      0.025     -0.043      0.966        -0.052     0.050
var_3          0.0098      0.025      0.393      0.695        -0.040     0.059
var_4          0.0253      0.030      0.856      0.394        -0.033     0.084
var_5          0.0160      0.027      0.596      0.553        -0.037     0.069
var_6         -0.0138      0.028     -0.486      0.628        -0.070     0.043
var_7          0.0100      0.024      0.418      0.677        -0.037     0.057
var_8         -0.0358      0.027     -1.335      0.185        -0.089     0.017
var_9          0.0180      0.026      0.707      0.482        -0.033     0.069
var_10        -0.0574      0.025     -2.288      0.024        -0.107    -0.008
==============================================================================
Omnibus:                        5.760   Durbin-Watson:                   1.815
Prob(Omnibus):                  0.056   Jarque-Bera (JB):                2.903
Skew:                          -0.147   Prob(JB):                        0.234
Kurtosis:                       2.219   Cond. No.                         1.72
==============================================================================


This is one of the reasons why $R^2$ is rarely used when doing model selection.

• I like the fact you took the time to give a worked example (+1). Suggestion: You could allude at the DoF associated with each model or perhaps nod towards an information criterion; eg. for the models you examine the AIC/BIC values indicate that model 1 is very possibly preferable to model 2. – usεr11852 Aug 3 '15 at 7:57