[SOUND] It sometimes happens that data that you want to analyze are not complete, there may be missing data. But what can you do then, time to panic? Of course, not, if you have enough data that you can analyze using an econometric model. You may use that model to make predictions for the missing data, and I used these predictions and treat them as original data. For example, look at the monthly consumer price index for Suriname observed for 2016 April, until August 2018. The observations for May and June of 2018 are missing due to a strike at statistics Suriname as you can see in the picture. The data are in the following table, as well as the data on fuel prices, and the exchange rate with the US dollar. The missing data are indicated by an A meaning, not available. Putting the three variables in one graphs gives you this picture. Now, what it is immediately obvious is that the three variables are trending upwards. And it may now be useful to use the exchange rate and the fuel prices of May 2018 and June 2018 to predict the missing consumer prices in those months. To do this we need an extension of the simple regression model where we include the paths of these variables, as well as the three variables at the same time. Such a model is called a vector autoregression, and it works like this. In the previous video I discussed a first-order autoregressive model for the casualties data. Without the trend, the model was given by this expression. Such a model is called a first-order autoregression with acronym AR(1). As the values of y depend on their own past, and here just one period ago. This model is very useful for forecasting into the future, as one can plug in the current observation on the right hand side of the equation to predict the next value. This AR(1) model is a univariate model as it makes yt dependent only on its own past, and not on the past of another variable. And when you want to do that, you may want to enlarge the model by introducing another variable xt as follows. The constant is set at zero to keep notation brief. And at the same time, you may have this, so yt depends on the past of yt and the past of xt. And at the same time, xt depends on its own past and the past of yt. The past of the two variables helps to forecast the future of both variables. The two equations together are called a vector autoregression of order 1, or in short VAR(1). The parameters in each of the two models can be estimated using the same least-squares technique as before. The VAR model can also be written in a so-called error correction format. This basically means that the left hand variable becomes a variable in differences, and the right hand side variables are in levels. Look again at this expression done by subtracting y t-1 on both sides, you get this, or even more convenient, this one here. Likewise, we can write the second equation as follows. And in this very special case that this restriction holds, we have a situation that is called co-integration or common trends. The VAR(1) model now has become a so-called error correction model as the two equations can be written like this. Where changes in the variables yt and xt are explained by deviations from a linear combinations of the level. That is y t-1- delta.x t-1, hence the word error correction. Back to the data for Suriname, if we would regress the consumer prices on the exchange rate and the fuel prices, we get this. Writing this as a linear combination like the following, we get the variable as in this figure. Which shows that this linear combination does not have a trend anymore like the individual variable seem to have. When this regression equation is used to predict the CPI of May and June 2018, we got 134.5 and 135.7 respectively. With the CPI values, we can compute the inflation rates also for these months. Plugging these two numbers in the table gives a full data set. And we can now estimate an error correction model for changes in consumer prices like this. Well, now also the changes in the exchange rate and the fuel prices are included. Least squares gives the following estimating results at the r-squared is 0.603. With this model for Suriname consumer prices, we conclude this video.