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# Quant Anly Pol Sci Ii POL 213

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This 26 page Class Notes was uploaded by Pierre Huel on Tuesday September 8, 2015. The Class Notes belongs to POL 213 at University of California - Davis taught by Bradford Jones in Fall. Since its upload, it has received 89 views. For similar materials see /class/187556/pol-213-university-of-california-davis in Political Science at University of California - Davis.

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Date Created: 09/08/15

Preliminaries Brad Jones1 1Department of Political Science University of California Davis April 15 2008 earch M elho ls Today Preliminaries Jon POL 213 Re arch Melholt Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Preliminaries gt Today Preliminary Concepts gt Most of which should seem familiar gt If not review regression text Jon POL 213 Re arch Melliolt Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Setting the Stage Basic Statistical Concepts gt Properties of probability V Classic definition implies long run relative frequency of some event A gt Bayesians tell us this is not always a good definition they39re right However let39s walk before we run V V PrA is real valued function defined on a sample space Important properties OgPrA 1 V A 1 PrABC1 2 PrABCPrAPrBPrC 3 v gt 2 implies exhaustive events 3 implies mutual exclusiveness Jones POL 213 Research Methods Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Basic Concepts gt Random Variable a real valued function defined on a sample space It39s an observable with two flavors Discrete and Continuous Probability Density Functions VVVV The PDF assigns probabilities to outcomes Jon POL 213 Re arch Melliolt Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Discrete Random Variable gt PMF for a DRV fX PrXxi V i12n 4 0 inxi 5 gt In words the probability that X is equal to some specific discrete value gt Die Rolls in R d1 lt samp1e1 6 100000 probrep16 6 replaceTRUE d2 lt samp1e1 6 100000 probrep16 6 replaceTRUE die roll lt d1 d2 histdie roll breaks seq0 12 by 1 probFALSE 1as1 colquotredquot mainquotDistribution of 100000 die rollsquot xlab quotValue of Rollquot Jones POL 213 Research Methods Bradford 5 Jane UC7Davis De t of Polit 39 PMF for 100000 die rolls 15000 10000 3 Er 9 LL 5000 0 l l x x x x x x x 0 2 4 6 8 10 12 Va ue of ROM rch M 39thods Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Discrete Random Variable gt PMF in R freqltcbindtabledie roll prva1ltfreq1000OO rolllt cbind2345678910 1112 plotroll prval colquotredquot mainquotProbabi1ity Mass Functionquot xlab quotValue of Rollquot ylabquotProbabi1ityquot Jones POL 213 Research Methods Bradford 5 Jones UC7Dav Artwork Probability Mass Function OM i Probabmty o Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Continuous Random Variable gt Continuous RVs have density functions gt The density is kind of like a smoothed out histogram gt The probability of any specific realization of X is assumed to be 0 Why gt we must integrate to define probability within an infinitesimally small differentiable area V fx in discrete case is easy to define in continuous case fx may take on a variety of forms The PDF fx for the standard normal V eixgZ V2w gt We use this distribution all the time z scores for example Jones POL 213 Research Methods fx 6 Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Continuous Random Variable gt The cumulative distribution function obtains probabilities FX PrX X X fxdx 7 Xmn gt Here fx is the PDF FX is the CDF gt In a sense the PDF is going to give us the height and the CDF gives us the area gt Note that it must be the case all area under the curve must integrate to 1 Fm Preoo X oo WW 1 8 gt Also important Fb7 Fa Pra X g b fab fxdx Remember this with ordinal logits and probitsl Jones POL 213 Research Methods Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Artwork um MW r n 2 um Swill1w mmva lunFALSE mmva 7 72 n 2 um Swill1w mmva lunFAL8E mmva 73 72 4 n Jone POL 21 Re earch Melllod Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Continuous Random Variable b b b b The classic linear model assumes y is continuous It may not be often will not be At what point regression fails us is a concern of this class With a binary dependent variable or categorical choice data regression Will fail us in certain kinds of ways to certain degrees gt But before we get to that let39s go on with a few more preliminaries Jones POL 213 Research Methods Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Sampling Distributions gt The sampling distribution of a statistic is the probability distribution of a statistic obtained from repeated sampling V Suppose our statistic is 0 V Let39s review some basic sampling concepts using R V Central Limit Theorem The sum of many independently and identically distributed random variables will tend to a normal distribution in the limit More specifically if the sum of independently and identically distributed variables has a mean u and a finite variance 02 then it will approximately follow a normal distribution gt To sustain statistical inference we rely heavily on the central limit theorem V lmportantly this result holds even if the population distribution is non normal Jones POL 213 Research Methods Let39s create a world of 100000 observations This is our population and this is what it looks like gt samplesizelt1000OO gt distltsamplergammasamplesize55 gt histdist colquotblue1quot gt meanX lt meandist meanX 1 0996132 Here the mean of the distribution is 100 Call this In The population distribution looks like this Jones POL 213 esearch M elho ls Bradford 5 Jones UC Davis Dept of Political Science Today Preliminaries Artwork Histogram of dist 10000 15000 x Frequency 5000 mm Jones POL 213 Research Methods some statistic from the population even if the population is nonnormal will tend to a normal distribution Suppose I took one sample of size 500 gt gt gt gt gt setseed510951 nsamp lt 1 res lt numericnsamp for i in 1znsamp resi lt meansampledist 100 replz meanres 1 1020171 The sample estimate call it 7 is 102 lt39s off from u Suppose we were to take 10000 samples of size 100 Jones POL 213 Research Methods Bradford 5 Jones UC Davis Dept of Political Science Today Preliminaries Histogram of res 1500 1000 Frequency 115 085 Jones POL 213 Research Methods Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Sampling Distribution This distribution is our sampling distribution It gives the distribution of means from repeated samples The mean of the sampling distribution is 100 The variability around the mean is the standard deviation VVVVV lnferentially the standard error reported to you in regression output is the standard deviation of the sampling distribution gt The problem of estimation is we only have one sample with which to work gt However the nice thing about the CLT is it gets us to the normal Or very close to it Jones POL 213 Research Methods Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Properties of Estimators gt Estimators have no inherent use to us without properties gt A random guess or mere dead reckoning is an estimator gt It39s just not very good gt Let39s review some properties of estimators Jon POL 213 Re arch Melliolt Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Estimators gt 9 is what we39re interested in gt This is a statistic gt Often we39re interested in the first moment of the sampling distribution gt That is the mean of the sampling distribution gt The question is what are the desirable properties of an estimator Jones POL 213 Research Methods Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Small Sample Properties Unbiasedness gt Small sample properties gt Unbiasedness 0 gt Equivalently 7 t9 0 gt The estimator is unbiased gt In contrast 7 t9 7 O gt lmplies biasedness in the estimator gt Note this property is a repeated sampling property Jones POL 213 Research Methods Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Small Sample Properties Efficiency gt Unbiasedness only tells us something about the central tendency of the sampling distribution gt An estimator is said to have minimum variance if var lt var Efficiency take two estimators and If each are unbiased but is a minimum variance estimator then is efficient or in words you may have used before best unbiased If is a linear function of sample data then is a linear estimator V V V Thus if is efficient and linear then in the class of linear estimators it is best unbiased You of course have seen this BLUE best linear unbiased estimator If the Gaussian assumptions hold the OLS estimator has this property V V Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Large Sample Properties gt Some estimators will not satisfy these properties in small samples gt Only in large samples do approximately equivalent properties hold gt These kinds of properties are asymptotic properties or large sample properties gt Asymptotic unbiasedness limrH00 E n 0 gt Asymptotic properties are directly tied to sample sizes small samples they will not hold gt You39ve seen this before V In small samples this estimator for the variance is biased in large samples the bias tends to 0 gt Thus the estimator is asymptotically unbiased Jones POL 213 Research Methods Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Large Sample Properties Consistency gt Consistency is a probabilistic statement nlergoPrl0i0llt6l 6gt0 9 gt Or plim 0 This is the consistency condition VV Note that unbiasedness can hold for any sample size consistency is purely an asymptotic property V A sufficient condition for consistency is that the bias and variance both tend toward 0 as n increases Note that the MSE criteria is not used in the OLS context because it is biased In large samples this bias diminishes V V The central limit theorem is an asymptotic theorem V That is asymptotic normality holds if the sampling distribution of 0 a N as the sample size increases Jones POL 213 Research Methods Bradford 5 Jones UCrDaviS Dept of Political Science Today Preliminaries Toward Likelihood gt Know your estimator and its properties gt Most all of the estimators from here on will only have large sample properties gt It therefore is very risky to apply models considered from here onward to small samples gt That doesn39t stop smart people from doing stupid things however gt Main point research design carefully conceived is incredibly important Jones POL 213 Research Methods

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