# 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