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Week 11 Review of statistical concepts Random variables- normally deal with continuous ones anywhere along con- tinuum e income- anything above £ Can be depicted on distribution functions x axis- continuous range over which variable X exists and Y is density func- tion Value taken between a and b is...

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Week 1 1.a Review of statistical concepts - Random variables- normally deal with continuous ones anywhere along continuum e.g income- anything above £0 - Can be depicted on distribution functions - x axis- continuous range over which variable X exists and Y is density function - Value taken between a and b is A - We have to integrate to find this- area under density function=1 - Distribution point F(a) represents the area under the density function up to the value a - Area to the left of a on density function= height of F(a) - Non decreasing function- probability goes up along the x axis

1.2 moment- mean and variance - Describe certain characteristics of a random variable and its density/ distribution function - Most common- mean and variance - Mean- central tendency of a density function of a variable X= mu or E(X)- expected value of x - Expected value- multiply the density of f(x) by x across the whole x axis - If symmetric would be centre point of distribution- helps us know unbiased - Whatever inside expected value- multiply that by f(x) - Variance- how dispersed var(x) find it by f(x) multiplied by x-mu all squared

1.3 Covariance and correlation - Mean and variable- property of single random variable - Covariance- how two variables X and Y are associated cov(X,Y): - Double integral- x-mu multiplied by y-mu multiplied by f(x,y) - Mu is with respect to which variable- different for x and y - Can be negative, positive or 0- intuitive- 0 no relationship

- e.g positive- more smokers, more lung cancer in that population- positive covariance with lung cancer and smokers - Correlation coefficient- two variables are positively or negatively related- and gives us strength of relationship - Corr(X,Y)= cov(X,Y)/sigmax times sigma y - Takes value between -1 and 1- closer to 0- weaker relationship - e.g correlation coefficient close to one between income and consumption - Correlation doesn’t imply causation- one may not cause the other

1.4 normal distribution - Symmetric bell shape- higher probability that x will take Value around E(X) - Lower probability taking a value in a tail - e.g heights income and exam results - 68% being within one standard deviation and 95% within 2 99.7% within 3 - Can transform any into a standard normal Z

1.5 random sampling - Collect sample- then find mean based off that - True unknown value- don’t have all info needed to work it out- estimate it - Need sample to be representative- estimator has good properties - Sample size n each xi drawn independently from the same distribution (population) therefore has same distribution - I.i.d- independently identically distributed

1.6 properties of estimators - x bar is an estimator of mu - If theatre hat is a good estimator- close estimate to the true value of theatre - Needs to be unbiased- if E(theatre hat)=theatre- mean are the same of the sample and the true mean - Distribution needs to be centred on the real mu- we can say it is unbiased- on average will yield the true value -

Efficiency- need as much of the estimator to be around the the point- variance must be small

- Estimator more precise if variance small- more likely to give the true value - Consistency- as size of sample increase- n to infinity- variance goes to 0good estimator- more accurate as n increases- will estimate true value across all - consistency means we could increase the sample size indefinitely and we would estimate the true value - Central limit theorem (CLT)- mean estimator is normally distributed with a mean of mu and a variance of sigma squared/n - As n increases variance tends to 0- sample mean is a consistent estimator - To say its efficient- comparison to show the variance is lower

1.7 hypothesis testing - Once we have estimated values for the parameter of interest- can test hypothesis about parameter - How likely our hypothesis is true given parameter estimate - Distinguish between value for x bar different from 25k as value of mu is different from 25k or even though it is 25k - Asking whether our estimated value is sufficiently far away from our hypothesised value to suggest that the hypothesis is incorrect or whether our estimate or sufficiently close to the hypothesised value to suggest the hypothesis is correct - Two sided hypothesis- either side of hypothesised value- can be more or less - We dont know variance of Xbar- use t when we dont know sigma - Can replace sigma squared with S squared- use a t dist. - always centred on 0- fatter tails than a normal - Area under=1 - Only has tn-1 if null hypothesis is true - If in tails- we use as evidence against the null. Would rejected the null. - Reject if absolute value of t stat is greater than critical value - Type 1 error- rejecting hypothesis then its true- alpha% of the time- size of a test probability of a type 1 error - Type 2 error- not rejecting a false hypothesis- denote it as beta- power of a test- not committing a type two error

- Type one is worse than two

Confidence bands - Provide an interval of values as an estimate of a parameter- have a certain level of confidence the true parameter lies between e.g 95% CI - Need to find critical values such that probability that the t stat is between minus tc and plus tc is 95%- contains mu - Tighter interval- less confidence trade off

Bivariate linear regression Discussion of the model - Bivariate econometric models- analysing simple models involving 2 variables - Want to reveal more about the relationship between 2 economic variables - Simple consumption function fits into this category- composes income and C - Use economic theory to indicate the causation of the relationship- changes in the value of X that cause changes in Y e.g - Usually indicate the explanatory variable by X dependent as Y - Causation runs from X to Y - Only interested in linear econometric models- straight line X and Y - How incomes effect their consumption e.g - More individual earns the more they can spend- positive relationship - Theory suggesting that variation in variable Y- depends linearly on variability in X- income in this example. - Y= B1+B2X- B1- y intercept B2 is the slope- how much Y changes when X changes - If we know these 2 B values- can know position of the line - Add some realism- add error term- donate as epsilon- e - Call this the population regression model- error term takes into account the randomness of human behaviour- prevents all income and consumption values being exactly on the line - May be above or below- e can be positive or negative - Treat e as random variable- accounts for random behaviour

- B2- importance- directly shows how X affects Y - dont have access to population to know these B values - Collect data on X and Y- estimate value for B1 and B2 - Data collection- representative- so can generalise the population- data cannot be biased- accurate data

- Use the line of best fit through the data points- best represents the trend X and Y Linearity: linear model if it has linear parameters- model parameters do not appear as exponents or products of the parameters If the model contains squares of products of variables- refer to it as linear regression if it is still linear in parameters Not linear- if theres a product of b- two bs multiplied together

Fitting the ‘best’ line - y=B1 + B2x + e - Unknown B1/2- don’t know position of the true population regression line - Collect good quality data and an estimation technique for b1/2 - Estimated b parameters- use a hat. - Issue for econometricians- how to get it - Issues- different model specifications- different techniques - Data collected not exactly in the form the model specifies- could affect the model and hence suggest a certain type of estimation prodecudure - Estimated regression line where B1 hat crosses the Y axis and B2 hat- slope - Error term e hat- called the residual- estimated version of the error term- distance each data point in the sample lies away from the estimate regression line

Mechanism for finding the best line - Residuals to be as small as possible- close to the line - Minimuse the sum of squared residuals - Any negative residuals become positive- dont cancel out with positive ones now

- Negative sloping line- would create a large value for S- not be a line chosen - Ordinary least squares (OLS) extremely common technique used

Ordinary least squares OLS -

OLS procedure minimises the sum of the squared residual

- To find differenfirst or-

the max or min of a functiontiate the function and set the der condition=0

-

- The Bottom two equations represent OLS estimators for B1/2 - Equations that involve the data observations on the variables X and Y - Can combine data with statistical technique to get estimates of the unknown parameters in the regression model - With these estimated values- can make statements about the economic relationship under the investigation

- B2 parameter interpreted as the MPC- income goes up by £1 consumption goes up by 78p - Estimators have sampling distributions- properties are important of these - Indicate whether the estimated values we obtain are likely to be representative of the true underlying parameter values

Properties of the OLS estimators - When using the OLS as a way of estimating regression parameters- interested in whether it will be unbiased efficient and consistent- will it be the best estimator - Unbiasedness- centre of the distributions - Efficiency- variance of the distributions- whether they are lower than other estimators - OLS- good properties of being best unbiased estimator- only happen if certain conditions satisfied- classic linear regression assumptions: - If these conditions are satisfied- best method of estimation- most accurate - Regression is linear in parameters and correctly specified - Regressor is assumed fixed- non stochastic (non random) in the sense that its values are fixed in repeated sampling. Strong assumption and implies that we can choose values of X in order to observe the effects on Y- more useful in experimental setting but if we relax this assumption and allow the regressor to be random- need the regressor to be independent of the error term - Expected value of epsilon=0 for all I- means that errors have a mean of 0 - Variance of error term is sigma squared, homoskedastic- variance of the error term is constant for all observations- No correlation between 2 error terms- not to be auto correlated - Each error term has the same normal distribution- same mean and variance- make this distributional assumption to enable us to serve the distributions of the OLS estimators and hence allow us to perform hypothesis tests. Not actually required for unbiasedness or efficiency - OLS will have the minimum variance- if assumptions hold.- Gauss-markov Hence OLS estimators B1/2 hat:

- Linear estimators- linear functions of the observations on Y - Unbiased- sampling dist of estimators are centred around the true but unknown values- on average estimators are correct - Efficient- smallest variances when compared to all other linear unbiased estimators- more likely to estimate a value close to the true value when estimator has a low variance

- Classical assumptions may be inappropriate- repercussions of properties OLS

Sampling distributions of the OLS estimators - Allow us to say something about the mean and variance of the estimator - Calc probabilities about the values mator might take - Allow us to create test stat and test about the population parameters we ing

that our estihypthess are estimat-

- Error

term normally distributed so are b1/2 hat

- Vari-

ance:

- Provide the lowest possible variance amongst the unbiased estimators - To get these variimpose some they didn’t holdlonger be the the best - Sampling dist of variate regres-

ance equations we needed to more classical assumptions- if variance obtained may no smallest thus OLS wouldn’t be the OLS estimator from a bision are therefore:

Coefficient of determination - Although method of OLS finds the best fitting line through scatter of data doesn’t necessarily its the good fitting line - To measure goodness of fit- how well fitted regression line fits through our scatter of observations- use coefficient of determination - R squared- equal to the squared correlation between variables x and y - Stat measures how much of the variation in the dependent variable is attributable to the regression model as opposed to the error

- Essentially-How much of the total sum of squares is attributed to the explained sum of squares rather than the residual (error) sum of squares - Regression line that fits well through points- residual variation small

- All data points lie exRSS=0- unlikely

actly on the line then

- R squared values tells total variation of Y is atgression line so that

us how much of the tributable to the re-

- R squared between 0 to 1- closer to 1 meaning better the fit of regression line - Closer to 0- worse the fit- higher the residual variation - Want it to be as close to 1 as possible- good fitting model

Hypothesis testing - B2=0.5 estimated b2 is unlikely to be exactly 0.5 even if hypothesis is true - We shouldn’t make simple inspections of the estimated value to decide whether a hypothesis is true - Test to distinguish between b2 that is different from 0.5 because the actual value of b2 is different from 0.5 T test: - Run a hypothesis test- need knowledge of the distributional aspects of OLS estimators -

Already said that b1 hat is normally dist with b1 mean and sigma etc

-

See slides

Test of significance - Used so often that stat software produce the test stat alongside the coefficient estimate for the B parameters - Called test of significance- null hypothesis is for a specific value that the parameter is = 0 - Useful because it can be used to test the significance of the X variable in the regression model Y=B1+B2Xi +Ei - Is variable important in determining dependent variable Y - Need to therefore see if B2=0 if it is 0- no influence on Y

Confbands:

- Numerator- subtract the null from it.

dence

Analysing reviews output

3 Multiple linear regression Disof

cussion the model

- Remodel in section 2- involves X and Y

gression

- Consumption- more than one independent variable

- K-1 explanatory variables and k parameters to estimator - B2- represents the intercept- B1 partial slope coefficients - B2- measures change in Y per unit change in X2- ceteris paribus

Ordinary least squares estimation - Still use OLS to estimate the unknown parameters in a multiple regression model- more than two parameters to estimate - Collect more data on all variables - Still minimise residualsthough

the sum of squared more complicated

- Need to find 3 b1-3, set them =0, gives:

first order conditions-

-

Properties of OLS - Procedure for deriving OLS estimators is the same here as in bivariate- Assumption CLRM2 needs modifying to hold for all explanatory variables in the model- independent of error term- across x1-xn - Need to add CLRM7- no exact collinearity exists between any of the explanatory variables. Means that there should not be an exact linear relationship between nay regressors - If they hold then OLS will be BLUE - Not estimating all beta parameters- cant assess the individuals effects of x2/3 on Y - Exact collinearity- perfect multicollinearity is often the result of practitioner error - Regressors can be highly correlated with each other- estimation issues-

Sampling dist of the OLS estimators - Sampling dist- hypotheses testing - Estimators normally dist- under the assumption of the normality of the disturbance term e - Therefore unbiased estimator- mean equals true mean - Estimate the error variance sigma squared= sum of errors squared/n-k where k is number of variables

Coefficient of determination - Measures of the goodness of fit- Helps choose between different models for the same dependent variable with different variables on the right - R squared=1-RSS/TSS - Problem- increases in value when you add more variables

- Artificially inflated- even if you’re adding insignificant variables- not important to the economic theory - If we choose model with highest r squared value- wont necessarily be the best - Should on r

be wary choosing between models based squared

- Adjusted

r squared- r bar squared

- Stat penalises the inclusion of irrelevant variables- more conclusive when comparing different models - Value only increases when extra variables added- have something important

Hypothesis testing - Restriction- the function of the parameters we specify under the null - When performing the test- effectively restricting the model parameters in the way specified in the null and checking where the data support the restriction - Restricted b2=0?- tell us if it impact Y - May be interesting in testing hypothesis involving single restriction but a restriction that contains multiple parameters - Whether two of the models regressors have the same impact on the dependent variable- vocational qualifications have same earnings potential as studying for a degree - Test multiple linear restrictions- joint tests - Test if b2=b5=1 - All of partial slope parameters=0 - any impact on Y - Jointly test linear restrictions e.g b3=b4 and b2=0 and B2+B5 - Testing single linear restrictions- F test and T - Test multiple linear restrictions- F test

Testgle -

ing sinlinear restrictions

Consider a k variable model- has k-1 explanatory variables and k parameters to estimate

- T test approach- calc t sion upon a comparison

stat and base our rejection decito t cv

- Degrees of freedom parameter n-k consistent with test having tn-2 - Estimate sigma squared done with= sum of epsilon squared/n-k

F test approach - Compares the stat RSS from the

residual sum of squares 2 models

- RSSr- residual sum of squares from the restricted model- model that the results from imposing the restrictions - RSSu- residual sum of squares from the unrestricted model- no restrictions imposed - Q- number of restrictions that we are imposing- this case 1 - N-k degrees of freedom from unrestricted model

- Compare f stat to f cv from its distribution - F stat always positive- check calculations if you get negative stat - Reject if test stat lies in tails

Testing single restrictions involving multiple parameters - T test approach- test linear combination of the parameters- e.g sum of parameters equals=1 or one parameters equals another

Testing multiple linear restrictions - Only use f test-

same equation for f stat

- Test overall signifiparameters

cance- all regression slope being=0

- Can use r squared nificance only for this

stat- if testing overall sig-

Difference between testing restrictions jointly and individually - Jointly- b1/2 simultaneously 0, individually- b2=0 rest are unrestricted - Individual t test- come up insignificant- both correlated- don’t show much on their individual tests - use joint one- reflects importance

Multicollinearity - Problem that occurs within multiple regressions - Explanatory related is exactly related to the other explanatory variable In the model - Assumption: Should be no perfect multicollinearity - consequence of which is that the software- refuse - Consequences for estimation with the presence of multi collinearity- amongst explanatory variables- still the best estimators - Multi collinear variables OLS unbiased still- repeated samp...