# Which regression and type of test to use?

I'm trying to find out how stocks and cash react (in terms of returns) when people feel bearish/bullish and when they move asset allocation between stocks/cash.

I have three data sets: monthly Dow jones closing prices, allocation of assets between cash and stocks, and monthly proportion(%) of bullish/bearish people. Would this be a linear regression as follows:

[Dji prices]$y_1$ + [cash]$y_1$ = $\alpha$ + [bearish %]$\beta_1$ + [bullish %]$\beta_2$ + [% allocation cash]$\beta_3$ + [% allocation stocks]$\beta_4$ + $e$

I am thinking to use the t-test and the null hypothesis being that $\beta_t$ = 0 where t=1, 2, 3, and 4. Does this mean I have to do 4 separate tests to test the significance of the four variables?

Is this the right approach to do a test to find the significance of each explanatory set?

• Is bearish/bullish a binary condition (where you're only ever one or the other)? Do the cash and stocks allocations add to 100%? – Glen_b Jan 26 '17 at 2:11
• @Glen_b bearish/bullish is not a binary condition but has another "condition" called neutral, which all add up to 100%. And yes, cash and stocks allocations sum up to 100% as well. – user112947 Jan 26 '17 at 3:05
• Then you can't identify coefficients for cash and stocks as well as an intercept, since cash + stocks is a constant – Glen_b Jan 26 '17 at 7:21

There is nothing wrong with doing 4 separate tests for each $\beta_i$.

As @MichaelChernick pointed out if you perform multiple hypetheses testing, the significance level for each comparison $\alpha_c$ has to be corrected to reduce type I error. Simplest correction is Bonferroni correction

$\alpha_c = \frac{\alpha}{k}$ where k - number of tests

Keep in mind that t-test is also used for one-sided hypothesis testing $\beta_j > 0$, thus you have to use $t_{1-\alpha_c/2}(n-p)$ quantile to test the hypoheses on $\alpha$ significance level.

You can also test multiple parameters at once $\beta_q = \beta_{q+1} = \dots = \beta_p = 0$ using the F-test which uses the F-Snedecor distribution

• Don't you think you should do p-value adjustments for multiplicity? – Michael R. Chernick Jan 25 '17 at 20:41
• @ŁukaszGrad Thanks. Is it correct to use [Dji prices]$y_1$ + [cash]$y_1$ as the dependent variables? I'm not sure if using the sum of the market prices and cash vectors are correct, and if I should be using market prices or returns. – user112947 Jan 25 '17 at 23:16
• @user112947 If i understand correctly, currently you simply estimate the sum (Dji prices + cash). If you want to estimate them separately, i.e. to have [Dji]y1 + [cash]y2 = XB + e , you can use multivariate linear regression. But then B is a matrix and you should use other tests (like likelihood ratio chi-squared test). Also since [alloc stock] + [alloc cash] = 1 you should drop one of them – Łukasz Grad Jan 26 '17 at 8:40