R Unbiasedness - Revision history

Rb: /* Analyse the distribution for estimated coefficients */

2016-02-02T22:43:55Z

‎Analyse the distribution for estimated coefficients

Rb: /* Analyse the distribution for estimated coefficients */

2016-02-02T22:43:16Z

‎Analyse the distribution for estimated coefficients

Rb: /* Analyse the distribution for estimated coefficients */

2016-02-02T22:42:54Z

‎Analyse the distribution for estimated coefficients

Rb at 22:41, 2 February 2016

2016-02-02T22:41:38Z

Rb at 22:38, 2 February 2016

2016-02-02T22:38:09Z

Rb: Created page with "This is R code to demonstrate that OLS parameter estimators are unbiased. # Code to demonstrate unbiasedness of OLS estimator Npop <- 100000 # Population Size N..."

2015-02-06T08:27:51Z

Created page with "This is R code to demonstrate that OLS parameter estimators are unbiased. # Code to demonstrate unbiasedness of OLS estimator Npop <- 100000 # Population Size N..."

New page

This is R code to demonstrate that OLS parameter estimators are unbiased.

# Code to demonstrate unbiasedness of OLS estimator

Npop <- 100000 # Population Size
Nsamp1 <- 100 # Sample Size

Q <- 300 # number of samples taken

u <- rnorm(Npop) # true error terms
beta <- matrix(c(0.5, 1.5),2,1) # true parameters
Xpop <- matrix(1,Npop,2) # initialise population X
Xpop[,2] <- floor(runif(Npop)*12+8) # exp var uniform r.v. in [8,19]
Ypop <- Xpop %*% beta + u # %*% is matrix multiplication

save_beta1 <- matrix(0,Q,1)

for (i in 1:Q ) {
sel1 <- ceiling(runif(Nsamp1)*Npop) # select Nsamp1 observations
Ysel <- Ypop[sel1,]
Xsel <- Xpop[sel1,]
reg_sel <- lm(Ysel~Xsel[,2]) # estimate regression in sample
save_beta1[i,1] <- reg_sel$coefficients[2] # save estimated beta_1

}

hist(save_beta1,breaks = 15, col = "blue",plot = TRUE)
# mean(save_beta1)

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	<pre class="r">hist(save_beta1,breaks = 15, col = "blue",plot = TRUE)</pre>		<pre class="r">hist(save_beta1,breaks = 15, col = "blue",plot = TRUE)</pre>
−	[[File:~~http://eclr.humanities.manchester.ac.uk/index.php/~~OLSunbiasedness_hist.jpeg\|frame\|none]]	+	[[File:OLSunbiasedness_hist.jpeg\|frame\|none]]

← Older revision		Revision as of 22:43, 2 February 2016
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	<pre class="r">hist(save_beta1,breaks = 15, col = "blue",plot = TRUE)</pre>		<pre class="r">hist(save_beta1,breaks = 15, col = "blue",plot = TRUE)</pre>
−	[[File:http://eclr.humanities.manchester.ac.uk/index.php/~~File:~~OLSunbiasedness_hist.jpeg\|frame\|none]]	+	[[File:http://eclr.humanities.manchester.ac.uk/index.php/OLSunbiasedness_hist.jpeg\|frame\|none]]

← Older revision		Revision as of 22:42, 2 February 2016
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	<pre class="r">hist(save_beta1,breaks = 15, col = "blue",plot = TRUE)</pre>		<pre class="r">hist(save_beta1,breaks = 15, col = "blue",plot = TRUE)</pre>
−	[[File:~~OLSunbiasedness_files~~/~~figure-html~~/OLSunbiasedness_hist.jpeg\|frame\|none]]	+	[[File:http://eclr.humanities.manchester.ac.uk/index.php/File:OLSunbiasedness_hist.jpeg\|frame\|none]]

← Older revision		Revision as of 22:41, 2 February 2016
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	<pre class="r">hist(save_beta1,breaks = 15, col = "blue",plot = TRUE)</pre>		<pre class="r">hist(save_beta1,breaks = 15, col = "blue",plot = TRUE)</pre>
−	[[File:OLSunbiasedness_files/figure-html/~~unnamed-chunk-6-1~~.~~png~~\|frame\|none]]	+	[[File:OLSunbiasedness_files/figure-html/OLSunbiasedness_hist.jpeg\|frame\|none]]

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-This is R code to demonstrate that OLS parameter estimators are unbiased.
+= OLS Estimator unbiasedness=
-    # Code to demonstrate unbiasedness of OLS estimator
+Here we will demonstrate that OLS estimators (assuming Gauss-Markov assumptions hold) are unbiased estimators.
-    Npop <- 100000  # Population Size
+We will first generate a large dataset (<code>Npop &lt;- 100000</code>) with <math>(y_i,x_i)</math> pairs of observations. The true relationship between the <math>x_i</math> and the <math>Y_i</math> is
-    Q <- 300  # number of samples taken
+<math>y_i = 0.5 + 1.5 x_i + u_i</math>
-    u <- rnorm(Npop)      # true error terms
+Then we will randomly draw <code>Q</code> samples (of sample size <code>Nsamp1</code>) from that population and estimate the regresison coefficients by OLS. Finally we will investigate the properties of the resulting distribution for the estimated <math>\hat{\beta}</math>.
-    save_beta1 <- matrix(0,Q,1)
+Of cours eyou should note that this is a rather artificial situation. In practice you will have one sample only and you will not know what the true coefficient is.
-    for (i in 1:Q ) {
+= Generate the population =
-    hist(save_beta1,breaks = 15, col = "blue",plot = TRUE)
+<pre class="r">   Npop &lt;- 100000  # Population Size
-    # mean(save_beta1)
+   u &lt;- rnorm(Npop)      # true error terms