The following code will attempt to replicate the results of the `numpy.linalg.lstsq()`

function in Numpy. For this exercise, we will be using a cross sectional data set provided by me in .csv format called “cdd.ny.csv”, that has monthly cooling degree data for New York state. The data is available here (File –> Download).

The OLS regression equation:

where a white noise error term. For this example the population-weighted Cooling Degree Days (CDD) (CDD.pop.weighted), and CDD measured at La Guardia airport (CDD.LGA). *Note: this is a meaningless regression used solely for illustrative purposes.*

Recall that the following matrix equation is used to calculate the vector of estimated coefficients of an OLS regression:

where the matrix of regressor data (the first column is all 1’s for the intercept), and the vector of the dependent variable data.

## Matrix operators in Numpy

`matrix()`

coerces an object into the matrix class.`.T`

transposes a matrix.`*`

or`dot(X,Y)`

is the operator for matrix multiplication (when matrices are 2-dimensional; see here).`.I`

takes the inverse of a matrix. Note: the matrix must be invertible.

## Back to OLS

The following code calculates the 2 x 1 matrix of coefficients, :

## load in required modules: import numpy as np import csv ## read data into a Numpy array df1 = csv.reader(open('/your/file/path/cdd.ny.csv', 'rb'),delimiter=',') b1 = np.array(list(df1))[1:,3:5].astype('float') nrow = b1.shape[0] intercept = np.ones( (nrow,1) ) b2 = b1[:,0].reshape(-1, 1) X = np.concatenate((intercept, b2), axis=1) Y = b1[:,1].T ## X and Y arrays must have the same number of columns for the matrix multiplication to work: print(X.shape) print(Y.shape) ## Use the equation above (X'X)^(-1)X'Y to calculate OLS coefficient estimates: bh = np.dot(np.linalg.inv(np.dot(X.T,X)),np.dot(X.T,Y)) print bh ## check your work with Numpy's built in OLS function: z,resid,rank,sigma = np.linalg.lstsq(X,Y) print(z)

## Calculating Standard Errors

To calculate the standard errors, you must first calculate the variance-covariance (VCV) matrix, as follows:

The VCV matrix will be a square k x k matrix. Standard errors for the estimated coefficients are found by taking the square root of the diagonal elements of the VCV matrix.

## Calculate vector of residuals res = as.matrix(women$weight-bh[1]-bh[2]*women$height) res = Y-(bh[0]+X[:,1]*bh[1]) ## Define n and k parameters n = nrow k = X.shape[1] ## Calculate Variance-Covariance Matrix VCV = np.true_divide(1,n-k)*np.dot(np.dot(res.T,res),np.linalg.inv(np.dot(X.T,X))) ## Standard errors of the estimated coefficients stderr = np.sqrt(np.diagonal(VCV))

Now you can check the results above using the `lm()`

function in R:

df1 = read.csv('/your/file/path/cdd.ny.csv',header=T) coef(lm(CDD.pop.weighted ~ CDD.LGA,data=df1)) ## (Intercept) CDD.LGA ## -7.6210191 0.5937734 summary(lm(CDD.pop.weighted ~ CDD.LGA,data=df1))