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| How we use the theory and data from economics, business, and social sciences, along with statistics to answer "how much" questions. |
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| Science of collecting, organizing, presenting, analyzing, and interpreting numerical data to assist in making more effective decisions |
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Some characteristic of a population of a sample.
(values vary across the population) |
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1. A random variable
2. A discrete Variable
3. A continuous variable |
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| Variable whose value is unknown until it is observed |
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| when a random variable can assume only a finite number of distinct values |
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| when a random variable can take an infinite number of possible values |
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| Levels of measurement of Data |
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Descriptive and inferential statistics influenced by the levels of data measurement
1. Nominal Data (Age, brand, sex)
2. Ordinal Data (Satisfaction Rating) |
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1. A measure of asymmetry in the data
2. A measure of Peakedness in the data |
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1. Arithmetic Mean
2. Geometric mean
3. Variance |
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1. Measure of location and shows the central value of the data
2. Uses multiplication instead of addition to summarize data
3. Variant or dispersion, spread around the mean |
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Measures relative variability in the data
(Standard Dev/MeanX100) |
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| Joint Probability Density Function |
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| Probability of two events occurring simultaneously |
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| Ability to obtain the probability distribution of an individual random variable |
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| Probability of the outcome X=2 given that Y=1 has occured |
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| The Chi Squared Distribution |
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Formed by summing the squared values of the independent standard normal variables
(as degrees of freedom get larger, distribution becomes more symmetrical)
(Deg of Freedom value determines the shape of the chi squared distribution) |
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| Variable formed by the ratio of two independent Chi squared random variables that have been divided by the degrees of freedom |
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For normal distributions, empirical rule tells us the proportion of the data we can expect surrounding the mean.
(For any set of observations, the minimum proportion of the values that lie within "K" standard deviations of the mean. K is any constant greater than 1) |
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| Assuming the population was normally distributed, sample means would also be normally distributed |
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| Best Linear Unbiased Estimator (BLUE) |
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| Estimator is unbiased, linear, has the smallest variance as "N" increases |
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| Range of values that may contain the true population mean |
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| Width of the Confidence interval depends on 3 things |
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1. Sample Size - Larger "N" means narrower
2. Level of Confidence - higher level returns wider interval
3. Variability - larger value of O returns a wider interval |
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| 5 Components of Hypothesis Testing |
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1. Null hypothesis denoted as Ho
2. Alternative hypothesis denoted as H1
3. A test Statistic
4. A level of significance/rejection region
5. Decision/Conclusion about hypothesis |
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| Computed based on knowledge of the relevant probability distribution for the variable under consideration |
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| Level of Significance/Rejection region |
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| rules set to decide whether you accept or reject the null |
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| One sample mean comparison test |
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| Test whether the sample mean value equals some mean value |
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| Rejection region if the decision rule the researcher adopts for rejecting or not rejecting the null (set %) |
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| Compare the computed "t" statistic w/ a critical value from the t tables w/ the relevant degrees of freedom and a particular level of significance |
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| Null hypothesis is true and we decide to reject it |
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| The null hypothesis is false and we decide not to reject it |
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1. Time Series form
2. Cross Section Form
3. Panel Data Form |
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1. data collected over discrete periods of time
2. data collected over different units at 1 point in time
3. data collected over sample units over period of time |
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1. Quantitative Data
2. Qualitative Data |
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1. Outcome expressed as numbers
2. outcome expressed as "either or" situations |
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Statistical process for estimating the relationships between variables
-Implies Y depends on X
-Line of best fit has smaller sum of squared residuals |
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| Assumptions in regression model |
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1. Regression model is correctly specified
2. regression model is linear in parameters
3. X is not random
4. Values of X vary across observations
5. Zero covariance between the error term and the X values |
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Null hypothesis w/ multiple conjectures, expressed w/ more than one equal sign
-Full model is called restricted model |
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| 3 Types of Log linear Models |
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1. Log-log Model- all variables are in log
2. Log-Linear model- Y variable in log, X variable not in log
3. Linear-Log model- Y variable not in log, X variable in log |
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| Let you estimate separate intercepts and slopes for different demographic groups |
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| Another type of specification that frequently occurs in economics |
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1. Multicollinearity
2. perfect Multicollinearity |
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1. When two or more explanatory variables are very highly correlated w/ each other
2. Cannot estimate all coefficients, individual coefficients have high standard errors |
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Involves observations on economic units of varying sizes
-As size of unit increases, more certainty associated with outcomes
-If errors don't have constant variance they are heteroskedastic |
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| One way to detect heteroskedasticity is to estimate your model using OLS, predict the residuals and examine graphs |
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| The White Test and Breusch Pagan Test |
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| Tests for continuous changes in variance |
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| Test for changes in the variance across discrete subgroups |
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| 5 Steps of the White Test |
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1. Regress Y variable by your explanators using OLS
2. Compute OLS residuals
3. Regress residual variable by a constant, all explanators, the squares of explanators, and all possible interactions between explanators
4. Compute R2 form step 3
5. Compare nR2 to the critical value from the Chi squared distribution with P degrees of freedom and decide if you should reject null |
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| Econometrician selects which explanators to include |
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| Steps in Goldfeld Quantd Test |
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1. Divide the N of observations into H group sizes
2. Choose two groups and form the null hypothesis
3. Estimate model for group 1
4. Estimate model for group 2
5. Relate the two groups
6. Compare G to the Critical value for the F statistic and if G is greater then reject the null |
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| Generalized Least Squares |
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Want to transform data so that it is homoskedastic, then we can apply OLS
-if nR2 is larger than the critical value we reject the null |
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| Refers to a sequence of observations over time |
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1. Static Model
2. Dynamic Model |
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1. A change in income at time (t) will cause a change in consumption at time (t) only
2. Past Effects the present, a change in consumption not only at time (t), but also in following periods |
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| Dynamic Models-Autoregressive Model |
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| Model where a variable is related to its past values |
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Third way dynamics can enter a regression relationship is the error term
-Variable is said to be autocorrelated or serial correlated if it exhibits correlation over time |
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| Testing for Serial Correlation |
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a test of serial correlation would be if the null hypoth. equaled 0 and the Alt hypoth. didn't equal 0
- If we reject the null then we are saying serial correlation exists |
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1. Estimate the model
2. Obtain the estimated residuals from the regression
3. Use the residual to compute the DW test statistic
-DW Test cannot accommodate time series model with lagged Y variables |
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| 2 Tests used for serial correlation |
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1. Durbin Watson - Focuses on first order serial correlation
2. Breusch Godfrey- takes into account correlations among disturbances lagged more than once |
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| Variance of error terms are not constant. Includes economic units of varying sizes |
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| Consequences of OLS estimator under Heteroskedasticity |
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OLS is unbiased
OLS is no longer efficient
Other linear estimators will have smaller variance
Estimated standard errors are incorrect
Confidence intervals will be incorrect |
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| Measures the relationship between the probability that an event occurs and its determinants |
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| checks to see if the time series variables have a unit root and if its data is stationary |
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| A stochastic trend that is a systematic pattern that is unpredictable |
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| 2 ways of measuring success in Logit Models |
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Count R2 as the measure of predicting if the even will occur.
Use Pseudo R2 as Log-Likelihood of the model |
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| Data obtained is not always convenient for presentation in a table in a regression |
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| Serial correlation Solutions |
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Newey-West procedure to correct estimated standard errors
Transform the data and use GLS if errors are AR(1).
Use Feasible GLS and estimate ρ using residuals |
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| convert the nonstationary series to its stationary counterpart by taking first differences |
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| either convert the nonstationary series to its stationary counterpart, or estimate a regression relationship that includes a trend variable |
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| any differences across economic agents can be captured by shifts in the intercept term of a standard OLS regression |
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| two-way fixed effects model |
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| equivalent dummy variable approach if there are time specific effects |
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| Using OLS to estimate a model in which the dependent variable is a binary variable |
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| Degree of similarity between a given time series and a lagged version of itself over successive time intervals |
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