Appunti Econometrics

Econometrics

1a lezione

U term is the error and is fundamental in econometrics. It is something which is omitted in the model specification that goes into this term. It is part, for example, of the wages, education, etc. In every model, we need to look at the error because it leads us to know that something wrong is in the model. On average, the error must be equal to 0. It's something completely random; if it's not completely random, there's something wrong in the model, and we need to change something until it reaches 0.

The error term represents all the factors that can affect someone's wage. It captures every wrong assumption. When our model is good enough, we can start to use it to test hypotheses and parameters. The econometric modelling is articulated in three steps: specification (writing the empirical model), estimate, and test. An example: yi = a + b'xi + ϵi. The variables in bold are explanatory variables. B is the vector of the estimator variable. Each x variable is unrelated to the error. If the explanatory variables are endogenous, it is not possible to use the OLS method. If there's a correlation with the error, we want to know the systematic relation.

The error must not be correlated; this changes the meaning according to the kind of data. This means the error in period T must be unrelated to the error of the previous period. Hypotheses errors have zero mean and are not correlated are condensed in: ~ i.i.d. (0, ϵ σϵ²). OLS is the most precise estimator, with the least variance. It is the linear unbiased estimator. The variance of the error must be constant, which is a homoscedasticity assumption. When the error is considered in a heteroscedasticity assumption, OLS is no longer the best estimator model, but we use GLS. If errors are correlated, neither OLS nor GLS are good methods.

2a lezione

In cross-sectional data, the order is not important; random sampling is reasonable for cross-sectional data. Explanatory variables, as well as independent variables, are random variables. DGP = data generating process. In identically distributed scenarios, the DGP is the same for every unit in our sample. Independently distributed, every unit has the same chance/probability to be extracted, in this case, cov is equal to 0. PDF = probability density function. Heteroskedasticity is one of the problems we must face with cross-sectional data.

Another type of data is given by time series data, consisting of observations on variables observed over a stretch of time (e.g., 2008, 2010, 2012...). Each observation is uncorrelated with other observations; however, it is not possible to assume that observations are independent. In time series data, we can observe a business cycle, which can go very well in some periods or even badly in others. They can also follow some index like GDP. We also have seasonality, for example, production increases during November/December and decreases during summer due to vacations.

If I want to analyze the movements in stock data, I need to have high-frequency observations and not annual observations such as can be with GDP. It is not possible to reshuffle the observations because we must know the temporal line to obtain and check whether there are cycles or not. The observations of the period T are correlated with the observations of the previous period. The assumption of identical distribution is substituted by the stationary assumption. In time series, it is also very important to check the autocorrelation analysis; another difference is in time series static model need to consider the temporal dependence of the variables. We have to take into consideration the learning by doing process.

Independently pooled cross sections are cross-sectional data at different points in time and pool them together. Panel data is a sample where you observe many units over time, and some units are followed over time; some units are not observed in the same period but join later.

Simple regression model

Two variables: dependent and explanatory combined in a linear model. In a normal distribution, skewness is equal to 0; skewness means that a distribution is symmetric. When it has a long tail on the right, it is positive; when it has a long tail on the left, it is negative. Kurtosis in a normal distribution is equal to 3; it is a measure of the probabilities under the tails. Together with the average, skewness, and kurtosis, we have a set of values; specifically, we have percentiles, ordered from the minimum to the maximum. The median is associated with the second quartile. Its specific value depends on how many observations we have in the sample; it is quite important because it is similar to the average. The median compared with the mean is not affected by perturbations in the tails.

It is a more robust index, a measure of central tendency; it has 50% of cases below and 50% of cases above. It is robust to the outlier. Comparing median and average, when the median is higher than the average, the distribution will have a bigger left tail; we will have extreme values tending to the negative, and the positive ones will be small numbers. Vice versa, when the mean is larger than the median, we will have more positive numbers. The perfect case is where the median is quite close to the average, so the skewness tends to 0, and the distribution is symmetric. The first quartile is associated with the first 25% of cases, a measure regarding the lowest number of our observations. The third quartile corresponds to the first 75% of cases, median is 50%.

Comparing the numbers of the 75% cases and the 25%, I obtain the interquartile range, a measure sometimes corresponding to the standard deviation. This range is a measure of variability similar to the SD, with the advantage that it is unaffected by the outliers. The outliers can inflate even more the SD rather than the mean. The comparison will tell us even more regarding the kurtosis; if the SD is higher than the interquartile, the kurtosis will be higher than 3, with many probabilities in the tail more than expected in a normal distribution.

The line inside the box is the median, the red line is the mean, and the part inside the box is the interquartile. In the variance, I can't use all the observations because I used all of them before obtaining the average. Since I have the average, I don't have to check all the observations. In the covariance, we have two variables compared; it can be positive if one increases and so does the other, or negative when one increases and the other decreases. The correlation is derived from the covariance. The variance can be considered as the covariance of a single term. The whiskers are important to understand whether you have outliers or not. If we don’t have mild outliers, we won’t have severe outliers.

3a lezione

Trim mean allows us to exclude extreme variables. The Jarque-Bera test is based on kurtosis; it is a test of normality. It allows us to know whether the life expectancy is distributed as a normal. It is a test dedicated to two hypotheses. A normal distribution has a skewness equal to 0, so the first hypothesis is to keep the skewness number equal to 0, the idea is the number we obtained is possible to be considered as 0. The other hypothesis is that the kurtosis must be equal to 3, from a statistical point of view. The JB test computes both kurtosis and skewness, and if kurtosis is equal to 3, and the skewness is equal to 0, our distribution is symmetric and the kurtosis is equal to 3.

Sktest variablename is used to test the skewness test. In the last two columns, we test the null in joint hypothesis if both are equal to 0 together; in this case, the two hypotheses together are rejected. Sometimes we have problems only in one or in both. We know that if the estimated probability is equal to or lower than 5%, we reject the null hypothesis. This test is based on a chi-square distribution; it is used when we want to test something that has to do with variances, standard deviations, and kurtosis. The t test is used for averages and the first moments, which can condense some characteristics; the second moment is the variance or the SD. When we want to test an average or a parameter, we use the student t test.

In this case, the chi-square distribution has 2 degrees of freedom because we are testing kurtosis and skewness. If we are on the left of the critical line, we are under the null; if we are outside of the red line, we do not accept the null. The third column compares the value with the chi-square critical value, and in this case, we reject the null. The critical value is a benchmark that we can know from the plot. This chi-square distribution is not under the null, so we must refuse it. The first possibility is to compare the computed value in adj2 chi2 with the critical value, and in this case, we are far away. The second possibility is that we know that the area is 5%, so the idea is that when we have to decide whether to accept or reject a hypothesis, what happens when we look at the computed value, the probability of making an error rejecting the null is 0, so if we reject the null, we’re not wrong.

In terms of the plot, our probability of making an error is equal to 0, so of course we reject the null. Also for the kurtosis, the probability of making an error is 0, whereas the probability of making an error with the skewness is 20%. If we reject the null, the probability of making an error is 20% and is higher than the kurtosis one.

• di invchi2tail(2, .05) allows us to obtain the critical value
• di chi2tail(2, 44.19) gives us the p-value

Why we want to know if it is normally distributed

• Outliers influence our distribution and the results.
• We are using linear models, so it is essential to know if the distribution fits over the assumption.

Reg estimate your model with the OLS; if I don’t have another term, the constant term is included in the definition. The constant term is estimated equal to the average, which is the last line of the results we obtained. But why average? A simple regression model is a model where we would like to explain the dependent variable with something. Simple regression model is Yi = α + βxi + ϵi. The OLS uses estimated values of hat, and they are the estimators. If we apply the formula to our data, we obtain the estimated.

In this case, the mean, which is the coef, is the estimator. β1 is the slope; the simple regression model is linear in the parameters; the error is quite important. In this term, all the omitted variables and mistakes we made writing a model are included.