# Why is a regression model of portfolio return giving smaller adjusted R-square (i.e., negative) than expected?

I have a question for adjusted $R^2$ given a specific regression model.

I am doing a project on January effect and I have a model from some journal using

$$R_i = a_0 + a_1D_{\mathrm{Jan}} + \varepsilon_i$$

where

• $R_i$ is daily return of portfolio/index,

• $a_0$ is non-January daily returns,

• $a_1$ is January returns over non January returns,

• $D_{\mathrm{Jan}}$ is dummy variable (1 for Jan, 0 otherwise).

With this model I tried to test whether January return is significantly greater than non-Jan returns, especially in small capitalization stocks.

So I have returns for size sorted portfolio 1 to 4, where Portfolio 1 (P1) consist of smallest cap stocks, P4 consist of largest cap stocks.

What I did was using regression program in Excel and SPSS, input all daily returns of the portfolio from Jan to Dec for 10 years as dependent variables and the dummy (1 for january and 0 for others) as an independent variable into the regression program and I get negative adjusted $R^2$ for all portfolios (P1-P4), mostly about -0.05. The value of $R^2$ itself was also very small at about 0.0001.

The journal I am basing my model on was using monthly returns as $R_i$, but I modified it into daily return. I still get negative adjusted $R^2$ when I use the monthly returns.

Can anyone please help me point out what was wrong in the model? Did I use the wrong input based on the given model? If yes, what should be the correct input? Or if the model was wrong, what model should I use?

Here are the results of my test, the p-value of the dummy variable and the adj $R^2$.

• For P1 0.54, adj $R^2$ -0.0002.

• P2 0.36, adj $R^2$ 0.0004.

• P3 0.68, Adj $R^2$ -0.0003,

• P4 0.14, adj $R^2$ 0.0005.

I understand that the variables are all insignificant. But I am confused in interpreting the $R^2$

• When you say project do you mean this is for school? If so you should change one of your tags to homework (probably the R one) Do you know that the adjusted R-squared should be something other than the value you're reporting (e.g. you're replicating a study which had a very different result)? Did you test the significance of the model itself? You may have done nothing wrong you may just have a model that is not significant given the data you used. Please provide more details. – Chris Simokat May 30 '11 at 19:12
• @chris: no, I did not test the significance of the model. I only tried to replicate the model from ir.lib.sfu.ca/retrieve/4269/etd2703.pdf page 44. But the paper doesn't give any R^2 value. Is it possible that the input for independent variable for regression running in excel was not only 1 or 0, but they are multiplied by something? When I tried to multiply them by diff between jan-non jan I got better R^2 at 0.11. But now the coeff size for a1 become too large at 0.9, while coeff a0 only 0.00005. – Ivana May 31 '11 at 16:58
• – landroni Feb 21 '16 at 8:50

## 1 Answer

You've included an interaction term without including both of the main effects that are the components of that interaction. According to standard practice, you need a term for January returns. Exceptions to this rule are rare though they have been discussed on this site recently at Including the interaction but not the main effects in a model

Beyond that (which may no longer apply after edits to the question), many people will obtain negative adjusted rsq when trying to predict something so difficult as stock returns. The rsq itself is so tiny that when the model gets penalized for its number of predictors (k), the resulting adjusted rsq will quite often go negative. This is especially true if the sample size is small, since N is, along with k, part of what determines the adjustment. Adjusted rsq = 1 - [(1 - rsq)(n - 1)] / (n - k - 1)

• P1 Coef Std Error P-value Intercept (ao) -0.00049 0.000163 0.002962 Djan (a1) 0.000353 0.000578 0.54113 Adj R Square -0.00026 P2 Coef Std Error P-value Intercept (ao) 0.000208 0.000235 0.377078 Djan (a1) 0.001853 0.000831 0.025876** Adj R Square 0.001619 P3 Coef Std Error P-value Intercept (ao) 0.000368 0.000206 0.074434 Djan (a1) 0.000298 0.000729 0.682311 Adj R Square -0.00034 P4 Coef Std Error P-value Intercept (ao) 0.000586 0.000292 0.045074 Djan (a1) -0.00152 0.001033 0.141486 Adj R Square 0.000475 – Ivana May 31 '11 at 2:37
• Sorry the last comment accidentally got uploaded before I edited it. – Ivana May 31 '11 at 3:02
• I did not test for the significant of the model, I only replicate them. They are based on ir.lib.sfu.ca/retrieve/4269/etd2703.pdf the model on page 44 and the result on page 47. The author did not explain anything about the R^2, but I was required to explain about the R^2. Since the result are weird, I dont know how to interpret them. I got only 1 more week to submit my result. my friend told me maybe the model was bad, should I use other model? (i will include p-value n adj R^2 in my question) – Ivana May 31 '11 at 3:32