noc21-hs39-lec29

Hello students, welcome back to the lecture series on Applied Statistics and Econometrics. So now let us have a look at today's agenda items. So as I said we are going to first talk about the restricted versus unrestricted model and there we are going to see an application of F-test, our good old friend F-test and then we are going to move on to an exciting area in econometrics and that is known as model misspecification issues. And there are several types of model misspecification issues in econometrics but in today's lecture we are going to talk about only one of them and that is called omitted variable bias and finally we are going to propose a solution to this problem as well. So first we are going to start with a formal definition of unrestricted and restricted models. So an unrestricted model is one model without any of the restriction imposed. So it contains all the variables that you may think that they may determine y, your dependent variable and then you know here comes the concept of restricted model. This is a kind of model on which the restrictions have been imposed. So for example you can say all regressors whose coefficients have been set to 0 they are called the excluded variables and if you have some excluded variables in one regression model then that leads to restricted model. So now note that restricted model works with fewer number of explanatory variables. So no wonder that for the case of restricted model the sum of square from error residuals is going to be a bit higher than the unrestricted model and we can also make similar observation for R square also. So as you know this slide is showing you here the unrestricted regression will always fit at least as well as the restricted one. So that means that unrestricted regression will have higher R square value than the restricted regression model because the sum of square errors from the restricted model is always greater than or equal to sum of square of errors residuals from the unrestricted model. So here as you see I am using the subscripts R and U you know for SSE symbol that you already know. So here the subscript R stands for the restricted model and the subscript U stands for the unrestricted model. Now the question is that how many restrictions one can put. So if you are in a multiple regression setup and if you have k number of explanatory variables maximum you can you know impose k you know number of restrictions. So here we are going to define a new notation and you know we are going to denote the number of restrictions imposed on a linear regression by Q. Now it is very much possible that each of these Q parameters that you are setting equal to 0 as you are you know putting some kind of restriction on the mother regression or the unrestricted model. You know they may be individually insignificant at some level of alpha but you know it you know may be the case that you know these Q parameters are jointly significant at you know the same level of significance. So we need a statistical testing and you know for that F test is going to be used. And in the next slide we are going to see how. So let us suppose that there are Q restrictions imposed and these are exclusion restrictions. So by that I mean that Q parameter values are set equal to 0 and H0 minor hypothesis states that you know Q which is you know I am assuming here to be strictly less than K you know have 0 coefficients and my alternative hypothesis says that H0 is not true. So that means that at least you know some of these exclusion restrictions are wrong. Now as I have shown you in the last slide that I expect a large increase in sum of square residuals or errors you know if I put restrictions on my mother regression or unrestricted model and you know if I finally end up getting a very large increase in SSE score then you know that is evidence against H0. So the first step here is to propose a test statistic and you know that I say F observed is my notation for that. And you know as I have shown you in the previous lecture that an F test could be conducted using two different metrics or measures. Of course you know one can make use of the sum of square error from the ANOVA table and you know the other way to do an F test is to make use of you know R square. So here I am going to propose two formula you know as you know they are going to you know be related to two different approaches that we have seen in the last lecture. So first we are going to talk about the R square measure. So now you know I am showing you the formula here. So R square U denotes the R square obtained from the linear regression model which is unrestricted. So that means there we do not have any exclusion restriction and then you know R square R actually means the R square obtained from the linear regression or OLS regression of the restricted model where I have imposed Q exclusion restrictions. And you know the difference needs to be divided by the degrees of freedom and you know that is basically Q and Q is what here, Q is basically the difference in the number of parameters between the unrestricted model and the restricted model. Now let us concentrate on the denominator of this same formula. So here 1-R square U is divided by the degrees of freedom and that is basically the degrees of freedom for the error term, right. You know if you remember the ANOVA table, the sources of variation will come from you know two angles and they are regression which is basically your regression model and the error. And for regression you have k number of you know explanatory variables, so you set k degrees of freedom and you know for the error source you set N-K which is basically the difference between number of observations and number of parameters to be estimated from the regression model as the degrees of freedom. So we are going to use the you know degrees of freedom from the error component from the ANOVA table. Now let us look at the second formula and here we are going to make use of the sum of square error directly. So again the subscript R stands for the restricted model and subscript U stands for the unrestricted model and the difference between the sum of squares from restricted and unrestricted model you know needs to be divided by the degrees of freedom Q and then you know this ratio needs to be further divided by sum of square error from the unrestricted model and that too divided by the degrees of freedom, okay. So now this F observed statistic you know either way we have calculated, it does not matter which formula we have chosen for, both formula will lead to this same theoretical result that F observed, my test statistic is now going to follow F distribution with 2 degrees of freedom and what are these 2 degrees of freedom? These are basically Q, the number of exclusion restrictions you have imposed on the mother regression model and the you know N-k-1, so the difference between the number of observations and the number of parameters to be estimated in the unrestricted model. Then you set your alpha and then you know this is the way you are going to make your decision, it is not new to you, so we have done F test previously, so you know you have to compare the value of the calculated test statistic which is F observed and if it is greater than the critical value that you find from F table for the degrees of freedom and the level of significance then you reject the null hypothesis and if you want to follow the p-value approach then of course you know you reject null hypothesis if p-value is less than alpha the set level of significance. And you know as I told you previously that a small p-value is enough evidence against you know H0 while a large p-value is not an evidence against the H0. Now we are going to look at these restricted versus unrestricted model case and you know how F test is going to help us to resolve this fight between you know restricted and unrestricted model through an example. So here in this slide I am going to talk about the restricted versus unrestricted model case from one demand function story. So if you remember couple of lectures back I have introduced the case of butter and margarine as you know 2 related goods, I was actually you know trying to get a demand function for butter consumption and then I showed you that you know margarine also plays a role because margarine is a you know substitute good for butter. Now if I you know want to go back to the same story of butter versus margarine and then you know how one can actually estimate a demand function for butter consumption. So suppose I am interested to feed a demand function for butter consumption and you know I want to you know define a log-log model for my demand function. Now we all know that demand or the purchase by consumers in the market is somewhat dependent on the advertisements made by the manufacturing company right and you know it is nowadays quite evident that you know people you know go by who is promoting that particular product and you know who is showing you know as the brand ambassador you know for a product. So these things matter. So now suppose there is a company which wants to statistically test whether expenses made on this advertisement you know be it any media it can be TV, it can be newspaper it does not matter. So you know the total advertisement expenditures made by a company how it affects the demand for you know the commodity of interest say butter okay. So in our demand function not only we will have the price of the main commodity which is butter and margarine the alternative or you know the substitute commodity from the point of the customer we are also going to add to potentially relevant variables and these are advertising expenses on you know butter and margarine. So now let us have a look at the final demand functional form that you know we are going to estimate. So as I said earlier we are going to estimate a log-log form. So you see you know my main variables which are basically the dependent variable and the explanatory variables which are coming from economic theory they are all logged. So logarithmic transformation has been used and now note that I also have you know a 5th variable that is time variable and for that the regression coefficient is beta 5. Now why do I need this extra variable which is not coming from microeconomic theory? It is because of the data. Suppose you know we have a data which is for several months or several quarters and then of course you know with time population changes, taste and preferences changes right. So there could be some annual change in the overall demand for you know any particular consumption item. So to capture the role of time we are you know adding this time variable in the regression because you know we collected data on all these variables over a substantially long period of time. Now we say that the model that you know I am showing you here you know under the first bullet point that is my full model or that is my unrestricted model. Now as a statistician I have this task at hand. So the company my client wants to know whether the advertising expenses on butter and margarine have impacted the demand for butter or not. So if you know they have come to me with this simple question how I am going to make use of this restricted versus unrestricted modeling framework to answer their question. So here I have to judiciously you know impose some exclusion criterion or restrictions on the mother model or the full model and what could be the exclusion restrictions here. So to develop the restricted model you know we can start with this presumption that butter and margarine advertising expenditures do not change the demand. So basically if you remember the previous slide when we were discussing the definition of restricted model I said that you know some parameters in the original regression model has to be set equal to 0 and what does that mean? That means that you know that particular explanatory variable for which you have set the parameter value equal to 0 then basically you are excluding that variable from the mother regression model. So you know if you do so then that means that the variable for which you are you know imposing this parameter value 0 restriction that is not linearly related with the dependent variable. So if advertising expenditures made by the company does not impact the demand for butter then basically I have to start with by setting the parameters for this advertising expense variables equal to 0 and that is what you know I am going to do here. So if you now concentrate again on the full or unrestricted model you see the coefficients for the advertisements on butter and advertisement expenditure on margarine are B3 and B4 respectively. So we are going to set null hypothesis which says that beta 3 equals to beta 4 equals to 0. So that is basically my exclusion restriction. So note that here I am imposing 2 exclusion restrictions. And what could be my alternative hypothesis? So alternative says that at least one of these beta 3 and beta 4 must be non-zero. Now I am going to you know show you the way how you can make use of ANOVA table or especially the you know information on sum of square error to conduct the F test. But you know you remember that you know I have also shown you the formula which involves R square formula. So similar technique or you know the steps that you know I am going to show you it can be followed for R square version of the formula or test as well. So now you concentrate on the ANOVA table. Suppose we have data on you know 51 time points, it could be 51 months, it could be 51 quarters, it does not matter and you know so here you know in the main regression the time variable actually takes you know 1, 2, 3 to 51 values, okay. So that is the way I am going to measure my time variable in the regression equation. Now you look at you know the slice of the ANOVA table. So here I am showing you 2 blocks of results. The first block talks about the full model or the unrestricted model and the second block talks about the restricted model and under both the models you know I am showing you the sources and the sum of squares you know for different components of total sum of squares, degrees of freedom and then mean square and then finally the F statistics. So you know if you look at the first block then I am not showing the total sum of squares because if you add the sum of squares for the regression component and the error component you are going to get the total sum of squares or SST. So I am skipping that. So anyway now let us you know look at the degrees of freedom column. So here you know you note that look at the mother model or the full model, the unrestricted model. Here we have 5 explanatory variables excluding the intercept term. Hence we have degrees of freedom equal to 5 and you know for the error component you know we have 45 degrees of freedom. How come? Because if you remember we have 51 number of observations and including the parameter we have 6 parameters to be estimated. So 51-6 you have 45. So now let us be back on the ANOVA table computation. So you divide the sum of squares number for each source or component by respective degrees of freedom to get the MS values and if you now divide the MS regression by MS error, so MSR divided by MSE you get the F value and you know this is the way you are going to actually get all these numbers that you are seeing here. And now note that as I have conducted 2 different regression experiments, I have got 2 sets of coefficients, 2 different values of R square and 2 different ANOVA tables. Now I am going to you know look at how I am going to combine the information from the restricted regression model and the full regression model into 1 formula which is my F observed test statistic and then you know conclude whether these restrictions are statistically significant or not. So now look at the 4th bullet point in the slide you know just you know right at the bottom of the table, there I am going to show you how you calculate the F statistic value. So here you know the formula I have shown you in the previous slide, you are just going to you know search for those particular entities in these 2 regression ANOVA tables and then you know if you plug those numbers back in the formula, you are going to an F statistic value that is going to be equal to 1.0838, okay. So once you get your you know calculated value of F statistic you know what to do, you set you know alpha level of significance say you know 5% and then you know you have to now look at the degrees of freedom that you have used in the calculation of the F statistic and then you have to go to the F table to find out the critical value. So what are the degrees of freedom here? So for the numerator the degrees of freedom is Q which is basically the number of exclusion restrictions you have imposed you know on the mother model or the full model. So here I am putting 2 restrictions because I am setting 0 value for 2 population parameters beta 3 and beta 4. So here the degrees of freedom is 2 and for the denominator I have these degrees of freedom capital N-capital K and that is going to be 45, okay. So now if you go to the F table then the critical value for these 2 degrees of freedom and alpha value 0.05 that value is 3.21 and now note that that is much, much higher than the calculated F statistic value which is 1.08. So you take a decision that we cannot reject H0, okay. So what do you mean by you know that you cannot reject H0? What is H0? H0 is basically talking about the exclusion restriction. So it is setting 0 value for the population parameters beta 3 and beta 4 and that actually tells you that jointly the advertising expenses on margarine and the butter commodities do not have any impact on the purchase or you know demand for butter in the market. And if you cannot reject null hypothesis then you say that you know your hypothesis is true you know with which you have started. So you know then you can tell your client that you know from my statistical analysis I have enough sample evidence to infer that the advertisement expenditures on margarine and butter you know they are not going to impact the demand for butter. Now we are going to move on to a new topic and you know that is called model misspecification. So I am saying that it is a new topic but it is not very different from the topic which you know we discussed last that is basically the case of hypothesis testing because here also you will see we are going to make use of this you know tool hypothesis testing to decide you know whether we have enough sample evidence you know so that we can infer something or not. So the inference you know that you know I am trying to draw here is not about you know a particular value of the particular population parameter. Here I am going to talk about you know drawing inferences about the correctness or the validity of the assumptions that I made when I you know framed my linear regression model. So you know in this lecture of course you know we are not going to talk about the entire you know span of this problem model misspecification. We are going to only talk about one type in the next lecture we are going to continue about that discussion. So let us first begin with you know a formal definition for model misspecification. So a model is said to be misspecified if the distribution of the data here by data I mean the you know variable under focus which is basically your dependent variable y. So if the distribution of data implied by the regression model under best possible parameterization differs from the distribution of data under the true generating process. So what do I mean in a nutshell you know about this quite bookish definition. So you remember I said that you know when you start the journey of building a classical linear regression model you have made you know certain assumptions about the data generation process and what do I mean by data generation process. So data generation process actually says that you know your y is a random variable and there is a process of you know generating values of y and what is this process from where it is coming. So when you you know write the classical linear regression model assumptions you first of assume that you know your explanatory variables are all fixed in nature. So you assume that your x's are not random. So there is no process if you know there is no randomness in the x's, x's are all you know exogenously given to you. So in the regression equation what is random for which you can expect that there could be a stochastic process. So that is basically the epsilon term, the stochastic disturbance term or the random error term whatever you want to call it that epsilon okay that you know is a random variable and as its random variable y will become a random variable too because you know your y now becomes a function of you know realization of you know this random variable epsilon. So there is a data generation process for y as well as you know you see it for epsilon. Now you start you know model building by making some assumptions about this you know random error. So remember the you know assumption of 0 conditional mean, then you know the assumption of homoscedasticity which is basically constant variance, then you know at last we have also said that you know one can assume normality as a you know distribution you know for the error term. So now these are all assumptions and you know you may be interested to know that you know in reality how far true these assumptions are. Note that you know a model is basically an abstraction from the reality. You know with your you know dreams and you know tool sets you know developing models by making assumptions and using you know these statistical techniques but you know real life is real life. You know it works you know in its own way and you know the assumptions that you are going to make to develop your regression model may not hold in reality. Data is data. You do not know whether it is going to obey the assumptions you are making you know while you are sitting in air conditioned room in front of a computer okay. So basically you know there has to be some mechanism to test whether the assumptions you are making to develop this classical linear regression models whether they are valid ones or not. And if somehow you figure out that the assumptions you made when you are developing your classical linear regression model you know they are invalid then actually they will lead to this model misspecification error. So why are we so much worried about misspecified models? What is the big deal if you know some of the assumptions that we made are not fulfilled by the data? You know the model misspecification errors can create havoc you know in your regression analysis and you know the inferences that you know you are going to draw out of your model can all be wrong you know depending on the magnitude of the misspecification error. So mostly they are going to create 2 types of nuisances. First you know they are going to create biases to the parameter estimates and what do we mean by bias? If you remember the discussions in part 1 of this course you know I you know introduced this term called bias. So bias is basically the difference between the true population parameter value and the mean of the estimator that you get from your you know OLS technique or you know whatever technique you are you know applying to you know draw some statistical inference from real life data. And you know the secondly the misspecified models are also going to create inconsistent standard errors. So inconsistent standard error means that you know if you you know get one sample you get one value of standard error and then if you move on to another sample you are going to get another value of standard error and you know if you are unlucky if there is you know model misspecification issue then you know they actually may be way apart from each other. So the dispersion of you know the coefficients that you know you are going to get from different samples is going to be very high and that is not good because you know that is going to impact your hypothesis testing like t-tests and all. Now there are many ways in which a regression model may be misspecified and here I am going to list down 5 of them. So first let us you know start with the case which is basically we are going to talk about in today's lecture. So if a statistically and theoretically relevant variable is not included in the regression equation then we will have the omitted variable problem. So that is basically the number one source of model misspecification. Then you know we are going to talk about the second source of model misspecification and that has to do with the functional form of the regression equation. So to start with you know we have assumed that my x the explanatory variable is you know linearly related to my dependent variable y and that is why you know we have a you know linear in parameter, linear in variable models but there could be nonlinearity you know involved and you know if you are you know simplifying a linear you know simplification may be too much of a simplification and then of course it will have its own consequences on the you know model estimates and the model fit and you know forcefully fit a linear regression equation then you know there is a functional form misspecification and that will lead to the second type of model misspecification. Now we move on to the third case and that is the measurement error in regressors. So here what do we mean by measurement errors in regressors? You know so far we have assumed that x's are given to you and x's are you know perfectly measured so there is no error in the value of x's but there could be you know error involved in the value of x's if they are not properly measured. Suppose you know I give you an example of say temperature and rainfall many, many times you know we use these variables in agricultural production function regression equations but you know in third world countries where the quality of tools and equipments are not very you know high and cutting edge you know there may be some inconsistency and errors while you know the weather stations are measuring rainfall and temperature. So in that case you know you may have some random noise you know embedded in the you know data on the temperature and rainfall. Now we are going to talk about the fourth source for model misspecification. So there could be unequal variability in y. So what do we mean by that? So remember when we you know started making assumptions for classical linear regression model I said that okay my errors you know are going to be homoscedastic in nature. So I am talking about constant variance. So if epsilon is not homoscedastic it implies that for you know ranges of data the variance you know for epsilon is now going to change you know that will have the similar effect on y as well because the distributions for y and epsilon they are linked to each other. So now if the homoscedasticity for epsilon breaks down then for y also we cannot expect constant variance for the entire range of y data. And you know homoscedasticity is very important assumption to make and you know if that breaks down then we will have a problem. Of course we are not going to you know discuss what sort of problem you know it can happen you know in this particular lecture. We save the discussion for the next lecture. And finally we come to the last point and that is basically saying that randomness in the regression linear model would not be independent from one time period to the next. So what do I mean by that? So actually you know I have camouflaged one term that you know I have already introduced in the course. Do you remember that you know I have introduced a term called serial correlation or autocorrelation? So by that I mean that when two error terms are correlated with each other across cross-sectional units or across time point units then actually we say that you know the randomness or the independence you know between the consecutive values of dependent variable breaks down and that is not good for you and that is what is basically being told in 0.5. So now we are going to talk about omitted variable bias through an example, okay. So let us go back to the you know microeconomic theory of demand function. So when we estimate a demand function, microeconomic theory tells us that the prices of complements and substitutes of the good you know are explanatory variables and they should be incorporated in the regression equation when you are trying to estimate a demand function for the commodity in the question. But I mean theory is theory, you know it is you know saying that you need to consider complementary goods and the substitute goods but you know does it provide the full list of complementary goods and substitute goods? Answer is no. So now which are the best complementary goods and you know which are the best substitute goods and you know how many of them you know you are going to keep in your demand equation you know that you know is you know not explained by theory. And also even if you make a list of say you know 3 complementary items and 4 substitute items for the commodity in question for which you want the demand equation to be estimated, you may not have data on you know this particular you know commodities. And if you do not have data then of course you cannot include that variable in the regression. So now we consider that you know there is a true population regression function which is you know perfect and you know here you know I am showing you the expression for that of course you know it is a simple you know version of the reality. Suppose I have only 3 explanatory variables and you know somehow you know for some reason I have omitted x3 variable. There could be reasons like you know I may be ignorant about the effect of that x3 variable on y or even if I know that you know there is an impact of x3 on y but I may not have data on it. So whatever be the reason from the regression equation that you know I am finally estimating from raw data, I do not have x3 in the regression equation. Now let us see the consequences. So now you actually end up estimating a regression equation involving 2 explanatory variables x1 and x2. Now note that if the x3 variable which you have omitted, if it is truly a relevant variable which actually has an effect on y, then you know the portion of y's variability that it can predict you know now that component will be you know now embedded in the random error that you have defined for your model. So now note that as b3 is not equal to 0, then b3 times x3 is also not going to be 0. So you know there is some you know non-zero component in the regression but you know there is no variable which is taking care of that. So it will go in the error term. So it will be part of the error term and you know the epsilon error term will be actually you know beta 3 times x3 plus u where u is basically new random error term which is you know obeying the standard assumptions for you know error terms that we make in classical linear regression model. So now you see you know how the omitted variable bias is you know emerging. So you take expectation of 2 both hands. So basically now expected value of epsilon you know will be you know sum of 2 components, expected value of beta 3 times x3 and expected value of u. Now u you know as I said that you know we can assume that it is following the standard assumptions for classical linear regression model. So expected value of u will be 0. So finally you are going to end up getting expected value of b3 times x3 and which is basically a non-zero number. So now you see the consequences you know like you know you do not have you know the error distribution which has 0 conditional mean. So the mean now could be positive, it could be negative but it is certainly not 0. So why this expected you know value of epsilon not being equal to 0 is a big deal because you know this comes very useful, this property you know comes very handy when you are trying to establish the unbiasedness of the OLS you know estimators. Now we are going to talk about another dimension of the problem. So if the excluded variable x3 happens to be linked with one of the existing regressors say x2, then you know epsilon is no longer independent of x2. And you know what does that mean? So your another assumption of classical linear regression model breaks down. So remember we said one assumption of exogeneity and there I you know have said this statistical criterion that you know covariance between my explanatory variables or the vector of explanatory variables you know should be not related with the vector of the error terms you know which is epsilon. So cov x e is not equal to 0. But if now you know x3 which you are missing is related with x2, then basically you know via the root of epsilon now you know it is going to be you know impact the coefficient of beta 2 as well. So you know if that is the case, if there is some correlation between x2 and x3, then what happens? Then you know x2 which is part of your linear regression equation that you are estimating will be correlated with the epsilon term or the error term. Why? Because that epsilon term now has you know that impact of that third variable x3 is hidden there. So in a nutshell, we can say that OLS estimators are biased and inconsistent. Now bias exists unless you know we are very lucky and when we could be extremely lucky so that you know although we have you know forgotten to incorporate one relevant variable but you know our regression results are not plagued by bias, you know for that we have to you know see two good things happening to us. So the first good thing is that the true coefficient should be equal to 0 and the included and the omitted variables are uncorrelated. Now in this slide, we are going to talk about omitted variable bias in a bit more detail. So you know the direction of the bias depends on the estimators as well as the covariance between the regressors and the omitted variables. Because you know I told you you know in the last slide only that suppose we are ignoring and omitting x3 and x2 and x3 are somehow you know correlated and then you know now your epsilon and x2 they are going to be correlated via this root of x3. So you see that the covariance between regressors you know those which are staying in the final regression equation and you know those which are omitted variables which are now part of your error, the covariance between these two are going to play a major role. So a positive covariance of the omitted variable with both you know an explanatory variable x and you know the dependent variable y will lead the OLS estimate of the included regressors coefficient to be greater than the true value of the coefficient. So basically there is an upward bias in this context. Now you know in this slide I am going to show you you know simple table which is going to you know talk about the nature of the bias. But you know in addition I should also tell you that it is very difficult to measure the extent of the bias. So let us have we you know have this simple table where we are talking about only two explanatory variables. So here you know I am showing you different possible cases. So let us you know look at you know different cells where you see this nature of the bias reported in color red. So let us talk about you know the you know cell 1 which is basically the northeast one. So here we see two conditions are given. So the first is saying that included and the omitted variables are positively correlated and not only that we also have to satisfy one more condition and that is given in the first row. So that is saying that omitted variable is positively correlated with dependent variable y. So only in that case you will get positive bias. Let us now you know going to look at one case of negative bias and then I will stop because rest you can follow. So now you know let us concentrate on the second column which is saying that included and the omitted variables are negatively correlated and now let us assume that my omitted variable is positively correlated with the dependent variable y which is given in first row. So again in that case you know you will see that bias will come but the direction will be different and this time we are going to get negative bias. Similarly you know you can see different combinations are going to lead to different types of bias. Now the bad part of you know omitted variable bias is that you know although we know there could be a problem but you know there is no hypothesis testing tool to help us so that we can you know detect that you know in our regression model whether we have a problem of omitted variable bias or not. So what to do? Well you know here you have to use your gray cell. So you have to you know do lots of thinking, you have to you know you know consult the previous literature, you have to figure out what wise men have you know talked about you know your particular regression problem that you are dealing with whether someone has you know suspected presence of you know omitted variable bias you know which is very relevant for the regression problem that you are currently handling. So from the literature you have to guess and also you have to you know use your intuition you know to figure out whether there is omitted variable bias or not in the regression equation that you know you are specifying. But suppose you know your intuition says that well there could be one variable which I am omitting or maybe the literature is suggesting you that you know the model specification that you currently have you know maybe you know suffering from omitted variable bias then how do you solve for it? So in the last slide of today's lecture we are going to discuss about it. So here as a potential solution econometricians have come up with you know a novel idea called instrumental variable and an instrumental variable is used in regression when you have endogenous variables. Now what do we mean by endogenous variables? Remember when we you know made the assumptions we said you know all our x variables are given so you know they are exogenous in nature. So basically you know a change in y will not have any impact on xs and you know if that breaks down you know we can say that you know our explanatory variables are endogenous in nature. So their values are actually determined within the model not outside the model okay. So the endogenous variables are basically the regressors which are you know influenced by other variables in the model. So as you know you see the values of the some of the explanatory variables may be determined within the model by other variables okay. Now suppose you know we have you know one regression model you know we are specifying a three explanatory variable linear regression model but there could be you know literature which is suggesting that you know well there could be another fourth variable which may have played a role in explaining the variation of y and it you know could have you know determined values of y but you know you are not you know incorporating it. Let that variable be denoted by x4. So now you know x4 is an omitted variable we are not you know taking care of that variable in the you know linear regression model to start with. Now you consider an alternative model where you have the first three explanatory variables but now you have one variable x star 4 for which you do not have any you know observations okay. So this x star 4 you can say that this is basically the you know omitted variable it is omitted that is why you know I am putting an asterisk you know on the top of x so that I can differentiate it from the other explanatory variables that actually are being used you know to get the coefficient estimates from raw data okay. So x star is basically you know the hidden impact of that omitted variable x4. Now you know we are going to make use of one trick. Suppose we do not have you know data on x star 4 even if we know that you know we are missing out on that variable but you know we just cannot you know get data on that particular variable. So what do you do? So you have to figure out you know some additional variable which is you know pretty much linked with you know the variable that you are missing. So basically you have to get a proxy variable for x star 4. And now let us assume that you know there is some variable x4 which is basically you know the you know proxy variable for x star 4. So basically x4 you can say it is an instrumental variable which is you know used as an instrument for the true omitted variable x4 and as they are doing the same job you know by explaining the variation of y you know they are denoted by the same symbol x4 okay. So now you know you see we have 2 equations 1 and 2. Now we are going to you know write the final model equation that we are going to estimate. So we write you know the mother regression model and now we are going to make use of this equation 2. So we are going to replace x star 4 from 1 by the expression in you know equation 2. So you know if we do so then you finally get an equation that is equation 3. Now note that we land up getting a completely new intercept term. So here if you you know look at the second line for the expression for y you know the new intercept term has become beta 0 plus beta 4 times delta 1 and you know I am you know introducing a new symbol to represent that you know in simple manner and that is alpha 1. And now also note that the coefficient for the instrumental variable or the proxy variable x4 is now you know product of 2 different coefficients, the coefficient of original you know omitted variable that is beta 4 and it is multiplied delta 2 which is establishing the linkage between the true omitted variable you know x star 4 and the proxy variable or instrumental variable x4. So that you know product is also you know being denoted by a new symbol alpha 4. So by estimating this model equation number 3 I mean we do not get the unbiased estimates for beta 0 and beta 4 but we get unbiased estimators for alpha 1, beta 2, beta 3 and alpha 4. Why are we so sure about that because now you see a new error term you know has been introduced in this model and that is basically W here if you can look at the equation number 3 again and note what is W? W is beta 4 times E plus epsilon. Now note that if you take expectation of Y then you know beta 4 times E that is going to be you know 0 because you know from equation 2 we can say that the you know E value, the E random error term is going to obey all the classical linear regression model properties and epsilon is anyway you know obeying the classical linear regression properties. So if you take expected value of W it is going to be 0. So we end our discussion on instrumental variables here. Next time we are going to come back with very nice case study on instrumental variables. Thank you. See you then. Bye.