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1 INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION This sequence introduces the principle of maximum likelihood estimation and illustrates it with some simple examples. p L

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2 Suppose that you have a normally-distributed random variable X with unknown population mean and standard deviation , and that you have a sample of two observations, 4 and 6. For the time being, we will assume that is equal to 1. p INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION L

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3 Suppose initially you consider the hypothesis = 3.5. Under this hypothesis the probability density at 4 would be and that at 6 would be p INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION L p(4) p(6)

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4 The joint probability density, shown in the bottom chart, is the product of these, p INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION L p(4) p(6) L

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5 Next consider the hypothesis = 4.0. Under this hypothesis the probability densities associated with the two observations are and , and the joint probability density is p INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION L p(4) p(6) L

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6 Under the hypothesis = 4.5, the probability densities are and , and the joint probability density is p INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION L p(4) p(6) L

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7 Under the hypothesis = 5.0, the probability densities are both and the joint probability density is p INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION L p(4) p(6) L

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8 Under the hypothesis = 5.5, the probability densities are and and the joint probability density is p INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION L p(4) p(6) L

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9 The complete joint density function for all values of has now been plotted in the lower diagram. We see that it peaks at = 5. p(4) p(6) L p L INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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10 Now we will look at the mathematics of the example. If X is normally distributed with mean and standard deviation , its density function is as shown. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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11 For the time being, we are assuming is equal to 1, so the density function simplifies to the second expression. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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12 Hence we obtain the probability densities for the observations where X = 4 and X = 6. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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13 The joint probability density for the two observations in the sample is just the product of their individual densities. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION joint density

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14 In maximum likelihood estimation we choose as our estimate of the value that gives us the greatest joint density for the observations in our sample. This value is associated with the greatest probability, or maximum likelihood, of obtaining the observations in the sample. joint density INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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15 In the graphical treatment we saw that this occurs when is equal to 5. We will prove this must be the case mathematically. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION p L p(4) p(6) L

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16 To do this, we treat the sample values X = 4 and X = 6 as given and we use the calculus to determine the value of that maximizes the expression. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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17 When it is regarded in this way, the expression is called the likelihood function for , given the sample observations 4 and 6. This is the meaning of L( | 4,6). INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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18 To maximize the expression, we could differentiate with respect to and set the result equal to 0. This would be a little laborious. Fortunately, we can simplify the problem with a trick. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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19 log L is a monotonically increasing function of L (meaning that log L increases if L increases and decreases if L decreases). INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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20 It follows that the value of which maximizes log L is the same as the one that maximizes L. As it so happens, it is easier to maximize log L with respect to than it is to maximize L. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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21 The logarithm of the product of the density functions can be decomposed as the sum of their logarithms. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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22 Using the product rule a second time, we can decompose each term as shown. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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23 INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION Now one of the basic rules for manipulating logarithms allows us to rewrite the second term as shown.

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24 INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION log e is equal to 1, another basic logarithm result. (Remember, as always, we are using natural logarithms, that is, logarithms to base e.)

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25 Hence the second term reduces to a simple quadratic in X. And so does the fourth. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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26 We will now choose so as to maximize this expression. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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27 Quadratic terms of the type in the expression can be expanded as shown. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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28 Thus we obtain the differential of the quadratic term. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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29 Applying this result, we obtain the differential of log L with respect to . (The first term in the expression for log L disappears completely since it is not a function of .) INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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30 Thus from the first order condition we confirm that 5 is the value of that maximizes the log-likelihood function, and hence the likelihood function. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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31 Note that a caret mark has been placed over , because we are now talking about the specific value of that maximizes the log-likelihood. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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32 Note also that the second differential of log L with respect to is –2. Since this is negative, we have found a maximum, not a minimum. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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33 We will generalize this result to a sample of n observations X 1,...,X n. The probability density for X i is given by the first line. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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34 The joint density function for a sample of n observations is the product of their individual densities. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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35 Now treating the sample values as fixed, we can re-interpret the joint density function as the likelihood function for , given this sample. We will find the value of that maximizes it. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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36 We will do this indirectly, as before, by maximizing log L with respect to . The logarithm decomposes as shown. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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37 We differentiate log L with respect to . INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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38 The first order condition for a minimum is that the differential be equal to zero. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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39 Thus we have demonstrated that the maximum likelihood estimator of is the sample mean. The second differential, –n, is negative, confirming that we have maximized log L. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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40 So far we have assumed that , the standard deviation of the distribution of X, is equal to 1. We will now relax this assumption and find the maximum likelihood estimator of it. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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41 We will illustrate the process graphically with the two-observation example, keeping fixed at 5. We will start with equal to 2. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION L p

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42 With equal to 2, the probability density is for both X = 4 and X = 6, and the joint density is INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION L p p(4) p(6) L

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43 Now try equal to 1. The individual densities are and so the joint density, , has increased. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION L p p(4) p(6) L

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44 Now try putting equal to 0.5. The individual densities have fallen and the joint density is only INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION L p p(4) p(6) L

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45 The joint density has now been plotted as a function of in the lower diagram. You can see that in this example it is greatest for equal to 1. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION L p p(4) p(6) L

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46 We will now look at this mathematically, starting with the probability density function for X given and . INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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47 The joint density function for the sample of n observations is given by the second line. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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48 As before, we can re-interpret this function as the likelihood function for and , given the sample of observations. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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49 We will find the values of and that maximize this function. We will do this indirectly by maximizing log L. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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50 We can decompose the logarithm as shown. To maximize it, we will set the partial derivatives with respect to and equal to zero. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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51 When differentiating with respect to , the first two terms disappear. We have already seen how to differentiate the other terms. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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52 Setting the first differential equal to 0, the maximum likelihood estimate of is the sample mean, as before. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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53 Next, we take the partial differential of the log-likelihood function with respect to . INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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54 Before doing so, it is convenient to rewrite the equation. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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55 The derivative of log with respect to is 1/ . The derivative of --2 is –2 --3. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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56 Setting the first derivative of log L to zero gives us a condition that must be satisfied by the maximum likelihood estimator. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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57 We have already demonstrated that the maximum likelihood estimator of is the sample mean. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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58 Hence the maximum likelihood estimator of the population variance is the mean square deviation of X. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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59 Note that it is biased. The unbiased estimator is obtained by dividing by n – 1, not n. INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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However it can be shown that the maximum likelihood estimator is asymptotically efficient, in the sense of having a smaller mean square error than the unbiased estimator in large samples. 60 INTRODUCTION TO MAXIMUM LIKELIHOOD ESTIMATION

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Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 10.6 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course 20 Elements of Econometrics

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