POS 3713 Study Guide Theories lead to hypotheses which lead to models We think the world works additively Y BetaX1 Beta2X o Some independent variables X1 and X2 cause changes in some dependent variables Y alpha betaX o Y is the dependent variable o X is the independent variable o Alpha is the constatnt o Beta is the coefficient A one unit change in x results in a beta unit change in the expected value of y o Alpha is also known as the y intercept o Beta is also known as the slope o Sat scores are supposed to estimate a students first year GPA GPA alpha beta times SAT o Your start with a scatter plot and draw a line that minimizes the sum of the squared errors o In y a bX a is our estimate of alpha and b is our estimate of beta o This method of minimizing the sum of the squared errors is called the ordinary least squared method o We may use OLS if our dependent variable is continous and our depedendt variable is normally distributed o The formulas both rely on the means of the variabes a single unusalaul value outlier can affect these stastics o since these are estimates based on a sample we are uncertain about the actual values of alpha and beta o We call our measure of uncertainity the standard error o There is a standard error for alpha and one for beta The smaller the standard error the more confident we are the our estimates are equal to the true values o The uncertainity around alpha is often important however for our purposes we are less concerned with alpha because it is not directly related to our hypothesis test o Alpha is important forfor calculating predicitoins of the dependent variable but that is not our goal o Our goal is to determine if the indepenent variable affects the dependent variable That s related to the uncerttainy around beta o Non directional hypotheses We use a two tailed test to check if the coeffectient is different from zero in eithert direction o Directional hypotheses we use a one tailed test to check if the coefficient is different from zero in the direction of our hypotheses T test t Beta 0 standard error of the coeefecient Most of the time the degrees of freedom will be n 2 o We can use predicted values to help understand the size fo the effects o Predicted value is the expected value of the dependent variable given the specified value of the independent variable Remember though that there is error o The model does not do a perfect job of predicting values of the dependent variable though it might get close in some cases o And it can tell us if the indepent variable is affecting the dependent variable population o Linearity a straight line adequately represents the relationship in the Fitting a linear model to a non linear relationship results in biased estimates o The values of the dependent variable are independent of each other The estimates are unbiased but the standard errors are typically biased downwards and because of this we are more likely to mistakenly reject the null hypotheses o OLS assumes the relationship between the independent variable and the depenendtn variable is linear o Leverage is when a case has an unusaual X value Leverage is not always bad o A case that has both an outlier and has leverage is said to influence the regression line It effects both the constant and the slope o A good prediction is not our goal the model could be a good test f our hypothesis even if it does a poor job predicting the dependent variable o The difference between the actual value and the predicted value is called the residual Ui Yi actual Yi o When we talk about the goodness of fit model we are talking about how well the model predicts the dependent variable The smaller the residuals the better the goodness of fit o The root mean squared error is a measure of the typical deviastions from the regression line MSE the square root of the sum of all the residuals squared n k K is equal to the number of parametersm most of the time 2 bivirate regression R 2 is the proportion of the variance in the dependent variable that our model explains It ranges from 0 1 The closer it is to 1 the more of the variation our model explains and the better our model is at predicting the dependent variable R 2 regression sum of squares total sum of squares Deviation from the mean predicted by our model over the toal devisations from the mean o Regression sum of squares toal sum of squares minus residual sum of squares Toal deviation from the mean minus deviations that our model does not explain o R 2 will get larger when you add more variables But this does not mean we should just add more variables to the model to increase the value of R 2 o The size of R 2 is most important we are trying to build the model that is the most predictive If we are simply hypothesis testing the value of R 2 is less important o In experemints both the treatment group and the control group should have similar charecteristsics o The typical political science study does not have random assignment It may not be ethical I e you cannot randomly assign religions o Since the groups are the same the changes are the result of treatment o Without random assignemtn we need to stastically control for potential confounding variables o Regression models only show correlation o To infer causation we must rule out alternative explanations o Spuriousness is there another factor your not considering Z x and z also y o Y alpha beta1X Beta2Z Y dependent variable X Independent variable Z controlling for spuriouslness o Including z in the model allows us to examine the effect of x holding z constant spuriousness Well know how x influences y without worrying about o In multiple regression model each beta tells us the partial effect of each different independent variable o Perform separate t tests to test for statsitcal significane of each indepenedent variable The key to note is that the beta in the bivirate regression model will be different from beta1 in the multiple regression model o Z is related to both X and Y Z explains some variation in X and some of the variation of Y o With more than two independent variables each beta is the effect of that particular independent variable holding al other indepenedent variables constant o The bivirate model has omitted variable bias That is our estimate of beta1 is incorrect because it includes the impact of z It likely overestimates the effect of the independent variable on the dependent variable With several independent variables you can calculate predicted values
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