# Introductory Example: Ordinary Least Squares (OLS) Estimation in Julia

* The script to reproduce the results of this tutorial in Julia is located here.

In this post, I show in Julia how to perform ordinary least squares (OLS) estimation after first simulating the OLS data generating process (DGP). At the end, we see that the parameter estimates converge to the true parameter as sample size grows large. If you have not yet installed Julia, it takes 5 minutes following these instructions.

As a reminder and to clarify notation, the OLS DGP is,

$Y = X\beta + \epsilon$,

where $Y$ is the $Nx1$ dependent variable, $X$ is the $N\mathrm{x}\tilde{K}$ matrix of independent variables, $\beta$ is the $\tilde{K}\mathrm{x}1$ vector of parameters that we wish to estimate, and $\epsilon$ is the $N\mathrm{x}1$ error satisfying $\epsilon \overset{\mathit{i.i.d.}}{\sim} \mathcal{N}\left(0,\sigma^2\right)$. Because we assume that the first column of $X$ is the constant number 1, we will find it useful below to work with $K \equiv \tilde{K}-1$. The least squares estimator is,

$\hat{\beta} = \left( X^T X \right)^{-1} \left( X^T Y \right)$.

Matrix Algebra and Simulated Random Variables: The OLS DGP

First, we generate the independent variables $X$, then we use $X$ to generate the independent variable $Y$. To begin, create a new file in Julia Studio and save it to your computer. In the file editor (script), insert the following commands:


using Distributions
N=1000
K=3
genX = MvNormal(eye(K))
X = rand(genX,N)
X = X'
X_noconstant = X
constant = ones(N)
X = [constant X]



The using command let Julia know that we will be using the Distributions package. The MvNormal() command initialized a multivariate normal distribution, which is an object including methods such as pdf for the probability distribution function and rand for drawing random variables.  eye(K) means $\mathit{I}_K$, the identity matrix of size $K\mathrm{x}K$. We only told MvNormal the covariance matrix, leaving the mean blank, which Julia assumes means that we would like a zero mean. The distribution of $X$ is arbitrary; we only used multivariate normal for simplicity. At the end, we concatenate the vector of ones to X using brackets, [].

Tip: To ensure that the matrices are of the appropriate dimension, use the size command. Above, we transposed $X$ after finding that rand returned it as a $3\mathrm{x}N$ matrix, when we need it to be $N \mathrm{x} 3$. Misaligned dimensions are one of the most common and frustrating errors to make in writing a program.

Now that we have created $X$ as a matrix containing a column of ones as well as three independent variables, we wish to multiply it by a vector of regression coefficients of length 4 (including the intercept) and add the normally distributed shock, $\epsilon$. For simplicity, we assume $\sigma^2=1$.

genEpsilon = Normal(0, 1)
epsilon = rand(genEpsilon,N)
trueParams = [0.1,0.5,-0.3,0.]
Y = X*trueParams + epsilon


Matrix algebra in Julia can be done as in a way comparable to Python, *(A,B), which means AxB, or in the more R-like way we used above, A*B. Then, you can click run (the little green arrow in Julia Studio) and it will perform the operations in the file above. To make sure it worked, you can now go to the Console and type,

julia> mean(Y)


and press Enter. If the code worked correctly, this should return the mean of simulated $Y$, which should be near the true intercept 0.1 (since each of the independent variables has mean zero, the true mean of $Y$ is just the intercept).

Functions in Julia: The OLS Estimator

Functions are defined in Julia using the command function, followed by the desired name of your function, and parentheses containing the arguments of the function. An end statement is required upon completion of the function definition. Indentation is required within the body of the function, and it is a good practice to explicitly include a return statement. The OLS estimator is defined with respect to any particular parameters as follows:

function OLSestimator(y,x)
estimate = inv(x'*x)*(x'*y)
return estimate
end


This function uses the dot product (*) three times, the transpose () twice, and the inverse (inv()) once. This function works for any matrices x and y with the appropriate dimensions. Once you have defined this function by running the file, you can obtain the OLS estimates of the true parameters by typing,

julia> estimates = OLSestimator(Y,X)


Because of the return statement, the parameter estimates will be returned by OLSestimator() and stored in estimates. If you compute estimates in the script, you can print them to the screen using,

println(estimates)


Finally, change the parameter defined at the beginning of your code. When N is small (say, 100), estimates will usually be further from trueParams than when N is large (say 10,000). This should be very easy to change; if you used the print statement for the estimates, just change the value of N, run the code, and see the new estimates printed to the console. In order to make your estimates reproducible (i.e., the exact same draws from the random distributions), set the random seed at the beginning of your code using,

srand(2)


where 2 is a possible seed you could choose.

Results

When I run the code above with random seed 2, I find that,

julia> estimates
4-element Array{Float64,1}:
0.11216
0.476437
-0.290574
0.0108337


so you can use this to check your results.

If you are only trying to estimate OLS, you can use the built-in command linreg(), but do not include the vector of ones, as linreg() will add another vector of ones. This is why I created X_noconstant above, which is just X without a column of ones. The syntax is,

julia> linreg(X_noconstant,Y)
4-element Array{Float64,1}:
0.11216
0.476437
-0.290574
0.0108337


and the estimates are identical to those in estimates.

1. Ken Bastiaensen says:

First of all, thanks for the effort to provide this website! Really nice.

In case you didn’t know, in Julia (as in Matlab) you can write the transpose as the single colon after a matrix variable, so X’ instead of transpose(X). I believe this makes inv(X’*X)*(X’*Y) more readable than the line in your code. Actually, it can be even shorter with just X\Y for the least square estimation.

Check out the introductory slides (#40 on OLS) from Douglas Bates:

And I know one should never argue on fonts, but may I suggest you use a fixed width font for the code snippets and use single spacing? I believe it would make these pages a bit more readable (but maybe it’s just me ;))

Good luck!
Cheers
Ken

1. Thanks for the tips! I’ll go back and switch the transpose commands to ‘.

Regarding double-spacing of the code snippets, I agree with you. I can’t seem to make WordPress accept both single-spacing and indentation. They have the “address” format for single-spacing, but it does not allow indentation; similarly for the Shift+Enter way of single-spacing. Let me look into it again.

All the best,