Running several regressions and storing all the data with regress

Dear MATLAB expers,

I'm trying to run many linear regressions each one consisting of one dependent variable and an independent variable, but I keep on stumbling upon the following error:

Unable to perform assignment because the left and right sides have a different number of elements.

Error in code (line 739)

tStat(i) = bint; % t-Statistic

It is an error concerning the size of 'tStat', 'Residuals', 'Outliers' and 'RegressStats' and I don't know how to fix it.

[NumRows, NumSeries] = size(stockReturns);
NumAssets = NumSeries - 2;

% Specifying start and end date of the calculation of alphas and betas
StartDate = datestr(stockReturnsDates(1));
EndDate = datestr(stockReturnsDates(end));
% 
Alpha = NaN(1, length(NumAssets));
Beta = NaN(1, length(NumAssets));
tStat = zeros(1818, 2, 2);
Residuals = NaN(length(NumAssets), 2);
Outliers = NaN(length(NumAssets), 2);
RegressStats = NaN(length(NumAssets), 2);

for i = 1:NumAssets
    
	% Set up separate asset data and design matrices
	DependentVariable = zeros(NumRows,1);
	IndependentVariable = zeros(NumRows,2);

    DependentVariable(:) = stockReturns(:,i) - stockReturns(:,1827); % Excess return of each asset
    IndependentVariable(:,1) = 1.0; % For calculating alpha
	IndependentVariable(:,2) = stockReturns(:,1826) - stockReturns(:,1827); % Excess market return

	% Estimate the CAPM for each asset separately.
    [b, bint, r, rint,stats] = regress(DependentVariable, IndependentVariable, 5);
    
     % Including values in matrixes
     Alpha(i) = b(1); % Intercept
 	 Beta(i) = b(2); % Betas
     tStat(i) = bint; % t-Statistic
  	 Residuals(i) = r; % Residuals
     Outliers(i) = rint; % Outliers
     RegressStats(i) = stats; % Includes R^2, F-stat and p-value
    
end

I would really appreciate your help since I've been stuck with this error for a while already.

[chiefjia]

I can't add the data as an attachment since it exceeds the allowed size

Answer 1

[Dave B]

You're getting this error because you're trying to store the result of bint (which is a 2 x 2 matrix) in a scalar location (tStat(i)). Note that bint is not the t-statistic, it's the upper and lower confidence intervals of your 2 predictors.

Maybe you wanted tStat(i,:,:) = bint(:)' (which seems to correspond to how you initialized the variable)? But I question your choice of name for this variable, and also would not recommend storing it in this shape of matrix (if anything consider 2 x 2 x 1818 (in which case tStat(:,:,i) = bint)...but what are you planning on doing with these values? It's likely more sensible to keep them in a 1818 x 4 matrix.

You'll run into the same problem with storing the remaining values: you need to specify a storage location that matches the size of the values you're going to store. r, rint, and stats will all be non-scalar. Consult the documentation to determine what size to expect, or run the regression on one column of your data to see what shape it will be. If you're really desparate and just want to store everything blindly without thinking about the size/shape, use a struct or cell array...though I'd discourage this as you're just postponing the inevitable problem of having to think about the size (and meaning) of the results of your regression.

A few other things I notice in your code:

• You're initializing all of your variables based on length(NumAssets) but you probably want to be using NumAssets (because length(NumAssets) is 1)
• You specified 5 as the third argument to regress. Looking at the documentation, the third argument (alpha) should be specified as a value between 0 and 1 (I'm thinking you wanted .05)

[chiefjia]

Dear Dave,

thanks a lot for your answer, your insights are really helpful!

I would like to ask you a few questions, since I still don't know which of your suggestions to adopt and how to adopt them. There should only be one predictor in the linear regression I implemented, the other value should be the intercept or alpha. The intercept should be the first column of the matrix 'Independent Variable' and beta should correspond to the second column of this same array. Have I written this in the right way in the code above?

Regarding the size of the output arrays: I've realised that the issue lies on the sizes of the two sides not matching, but I just don't know what size I should set for each one of the output variables. The code works if I do as you said and set 2x2x1818 for the size of tStat and do tStat(:, :, i), but I want to later match every single value of the output variables with the predictor and intercept, which feels more complicated with a 3 dimensional matrix than it would be with a 2 dimensional one. I have tried to set it up as you said (1818x4), but I keep on getting the error. I'm trying to do the aforementioned as explained below:

tStat = zeros(4, NumAssets);
tStat(:, i) = bint; % Confidence intervals

Could you also make suggestions on how to rearrange the sizes of the output variables so that I can store all the data to make it easily understandable and match it with each one of the regressions?

Thanks a lot for your insights, as you have noticed I'm still a beginner in the programming world and MATLAB :)

[Dave B]

@chiefjia - yes I think you have set up your independent and dependent variables appropriately for slope and intercept. When you are unsure in these case, the best way to find out is to explore with some simple test data where you know what it will look like, and I generally like to use plots to visualize:

x=(1:100)';
y=10+x+randn(size(x))*5;

b = regress(y, [ones(size(x)) x]);
xi = [1 100];
yi = b(1) + xi.*b(2);

scatter(x,y,'.')
hold on
plot(xi,yi)

Now you can see that the use of regress is as you expect, and you can also interrogate all of the output arguments and make sure you understand them (if you like).

I agree that storing this in three dimensions is going to be a pain. Your code snippet would work (as in it will run) if you added a (:)

tStat = zeros(4, NumAssets);
tStat(:, i) = bint(:); % Confidence intervals

The (:) says turn this into a column vector regardless of its shape. Each colum of tStat would correspond to the columns of bint appended to eachother. That's a little weird, because it means it's intercept-low, slope-low, intercept-high, slope-high. Let's see that in the example from above:

[b, bint] = regress(y, [ones(size(x)) x]);
bint

bint = 2×2

10.0879 13.4807 0.9419 1.0003

bint(:)

ans = 4×1

10.0879 0.9419 13.4807 1.0003

That's a fine way to store it, but I personally wouldn't like it. For me it's much more natural to keep the CIs together:

bint([1 3 2 4])'

ans = 4×1

10.0879 13.4807 0.9419 1.0003

[chiefjia]

Dear Dave,

thanks a lot for your feedback, I really appreciate it! I know get a better understanding of how this function works and the use of parentheses and colons :)