Panel Data Models. Chapter 5. Financial Econometrics. Michael Hauser WS15/16
|
|
- Gabriel York
- 7 years ago
- Views:
Transcription
1 1 / 63 Panel Data Models Chapter 5 Financial Econometrics Michael Hauser WS15/16
2 2 / 63 Content Data structures: Times series, cross sectional, panel data, pooled data Static linear panel data models: fixed effects, random effects, estimation, testing Dynamic panel data models: estimation
3 Data structures 3 / 63
4 4 / 63 Data structures We distinguish the following data structures Time series data: {xt, t = 1,..., T }, univariate series, e.g. a price series: Its path over time is modeled. The path may also depend on third variables. Multivariate, e.g. several price series: Their individual as well as their common dynamics is modeled. Third variables may be included. Cross sectional data are observed at a single point of time for several individuals, countries, assets, etc., x i, i = 1,..., N. The interest lies in modeling the distinction of single individuals, the heterogeneity across individuals.
5 5 / 63 Data structures: Pooling Pooling data refers to two or more independent data sets of the same type. Pooled time series: We observe e.g. return series of several sectors, which are assumed to be independent of each other, together with explanatory variables. The number of sectors, N, is usually small. Observations are viewed as repeated measures at each point of time. So parameters can be estimated with higher precision due to an increased sample size.
6 6 / 63 Data structures: Pooling Pooled cross sections: Mostly these type of data arise in surveys, where people are asked about e.g. their attitudes to political parties. This survey is repeated, T times, before elections every week. T is usually small. So we have several cross sections, but the persons asked are chosen randomly. Hardly any person of one cross section is member of another one. The cross sections are independent. Only overall questions can be answered, like the attitudes within males or women, but no individual (even anonymous) paths can be identified.
7 7 / 63 Data structures: Panel data A panel data set (also longitudinal data) has both a cross-sectional and a time series dimension, where all cross section units are observed during the whole time period. x it, i = 1,..., N, t = 1,..., T. T is usually small. We can distinguish between balanced and unbalanced panels. Example for a balanced panel: The Mirkozensus in Austria is a household, hh, survey, with the same size of each quarter. Each hh has to record its consumption expenditures for 5 quarters. So each quarter 4500 members enter/leave the Mikrozensus. This is a balanced panel.
8 8 / 63 Data structures: Panel data A special case of a balance panel is a fixed panel. Here we require that all individuals are present in all periods. An unbalanced panel is one where individuals are observed a different number of times, e.g. because of missing values. We are concerned only with balanced panels. In general panel data models are more efficient than pooling cross-sections, since the observation of one individual for several periods reduces the variance compared to repeated random selections of individuals.
9 9 / 63 Pooling time series: estimation We consider T relatively large, N small. y it = α + β x it + u it In case of heteroscedastic errors, σi 2 σ 2 (= σu), 2 individuals with large errors will dominate the fit. A correction is necessary. It is similar to a GLS and can be performed in 2 steps. First estimate under assumption of const variance for each indiv i and calculate the individual residual variances, s 2 i. s 2 i = 1 T (y it a b x it ) 2 t
10 10 / 63 Pooling time series: estimation Secondly, normalize the data with s i and estimate (y it /s i ) = α (1/s i ) + β (x it /s i ) + ũ it ũ it = u it /s i has (possibly) the required constant variance, is homoscedastic. Remark: V(ũ it ) = V(u it /s i ) 1 Dummies may be used for different cross sectional intercepts.
11 Panel data modeling 11 / 63
12 12 / 63 Example Say, we observe the weekly returns of 1000 stocks in two consecutive weeks. The pooling model is appropriate, if the stocks are chosen randomly in each period. The panel model applies, if the same stocks are observed in both periods. We could ask the question, what are the characteristics of stocks with high/low returns in general. For panel models we could further analyze, whether a stock with high/low return in the first period also has a high/low return in the second.
13 13 / 63 Panel data model The standard static model with i = 1,..., N, t = 1,..., T is y it = β 0 + x it β + ϵ it x it is a K -dimensional vector of explanatory variables, without a const term. β 0, the intercept, is independent of i and t. β, a (K 1) vector, the slopes, is independent of i and t. ϵ it, the error, varies over i and t. Individual characteristics (which do not vary over time), z i, may be included y it = β 0 + x it β 1 + z i β 2 + ϵ it
14 14 / 63 Two problems: endogeneity and autocorr in the errors Consistency/exogeneity: Assuming iid errors and applying OLS we get consistent estimates, if E(ϵ it ) = 0 and E(x it ϵ it ) = 0, if the x it are weakly exogenous. Autocorrelation in the errors: Since individual i is repeatedly observed (contrary to pooled data) with s t is very likely. Then, Corr(ϵ i,s, ϵ i,t ) 0 standard errors are misleading (similar to autocorr residuals), OLS is inefficient (cp. GLS).
15 15 / 63 Common solution for individual unobserved heterogeneity Unobserved (const) individual factors, i.e. if not all z i variables are available, may be captured by α i. E.g. we decompose ϵ it in ϵ it = α i + u it with u it iid(0, σ 2 u) u it has mean 0, is homoscedastic and not serially correlated. In this decomposition all individual characteristics - including all observed, z i β 2, as well as all unobserved ones, which do not vary over time - are summarized in the α i s. We distinguish fixed effects (FE), and random effects (RE) models.
16 16 / 63 Fixed effects model, FE Fixed effects model, FE: α i are individual intercepts (fixed for given N). y it = α i + x it β + u it No overall intercept is (usually) included in the model. Under FE, consistency does not require, that the individual intercepts (whose coefficients are the α i s) and x it are uncorrelated. Only E(x it u it ) = 0 must hold. There are N 1 additional parameters for capturing the individual heteroscedasticity.
17 17 / 63 Random effects model, RE Random effects model, RE: α i iid(0, σ 2 α) y it = β 0 + x it β + α i + u it, u it iid(0, σ 2 u) The α i s are rvs with the same variance. The value α i is specific for individual i. The α s of different indivs are independent, have a mean of zero, and their distribution is assumed to be not too far away from normality. The overall mean is captured in β 0. α i is time invariant and homoscedastic across individuals. There is only one additional parameter σ 2 α. Only α i contributes to Corr(ϵ i,s, ϵ i,t ). α i determines both ϵ i,s and ϵ i,t.
18 18 / 63 RE some discussion Consistency: As long as E[x it ϵ it ] = E[x it (α i + u it )] = 0, i.e. x it are uncorrelated with α i and u it, the explanatory vars are exogenous, the estimates are consistent. There are relevant cases where this exogeneity assumption is likely to be violated: E.g. when modeling investment decisions the firm specific heteroscedasticity α i might correlate with (the explanatory variable of) the cost of capital of firm i. The resulting inconsistency can be avoided by considering a FE model instead. Estimation: The model can be estimated by (feasible) GLS which is in general more efficient than OLS.
19 19 / 63 The static linear model the fixed effects model 3 Estimators: Least square dummy variable estimator, LSDV Within estimator, FE First difference estimator, FD
20 20 / 63 [LSDV] Fixed effects model: LSDV estimator We can write the FE model using N dummy vars indicating the individuals. y it = N j=1 α j d j it + x it β + u it u it iid(0, σ 2 u) with dummies d j, where d j it = 1 if i = j, and 0 else. The parameters can be estimated by OLS. The implied estimator for β is called the LS dummy variable estimator, LSDV. Instead of exploding computer storage by increasing the number of dummy variables for large N the within estimator is used.
21 21 / 63 [LSDV] Testing the significance of the group effects Apart from t-tests for single α i (which are hardly used) we can test, whether the indivs have the same intercepts wrt some have different intercepts by an F -test. The pooled model (all intercepts are restricted to be the same), H 0, is y it = β 0 + x it β + u it the fixed effects model (intercepts may be different, are unrestricted), H A, y it = α i + x it β + u it i = 1,..., N The F ratio for comparing the pooled with the FE model is F N 1,N T N K = (R2 LSDV R2 Pooled )/(N 1) (1 RLSDV 2 )/(N T N K )
22 22 / 63 [FE] Within transformation, within estimator The FE estimator for β is obtained, if we use the deviations from the individual means as variables. The model in individual means is with ȳ i = t y it/t and ᾱ i = α i, ū i = 0 ȳ i = α i + x i β + ū i Subtraction from y it = α i + x it β + u it gives y it ȳ i = (x it x i ) β + (u it ū i ) where the intercepts vanish. Here the deviation of y it from ȳ i is explained (not the difference between different individuals, ȳ i and ȳ j ). The estimator for β is called the within or FE estimator. Within refers to the variability (over time) among observations of individual i.
23 23 / 63 [FE] Within/FE estimator ˆβ FE The within/fe estimator is ( ) 1 ˆβ FE = (x it x i )(x it x i ) (x it x i )(y it ȳ i ) i t This expression is identical to the well known formula ˆβ = (X X) 1 X y for N (demeaned wrt individual i) data with T repeated observations. i t
24 24 / 63 [FE] Finite samples properties of ˆβ FE Finite samples: Unbiasedness: if all x it are independent of all u js, strictly exogenous. Normality: if in addition u it is normal.
25 25 / 63 [FE] Asymptotic samples properties of ˆβ FE Asymptotically: Consistency wrt N, T fix: if E[(x it x i )u it ] = 0. I.e. both x it and x i are uncorrelated with the error, u it. This (as x i = x i,t /T ) implies that x it is strictly exogenous: E[x is u it ] = 0 for all s, t x it has not to depend on current, past or future values of the error term of individual i. This excludes lagged dependent vars any xit, which depends on the history of y Even large N do not mitigate these violations. Asymptotic normality: under consistency and weak conditions on u.
26 26 / 63 [FE] Properties of the N intercepts, ˆα i, and V( ˆβ FE ) Consistency of ˆα i wrt T : if E[(x it x i )u it ] = 0. There is no convergence wrt to N, even if N gets large. Cp. ȳ i = (1/T ) t y it. The estimates for the N intercepts, ˆα i, are simply ˆα i = ȳ i x i ˆβ FE Reliable estimates for V( ˆβ FE ) are obtained from the LSDV model. A consistent estimate for σu 2 is ˆσ u 2 1 = ûit 2 N T N K with û it = y it ˆα i x it ˆβ FE i t
27 27 / 63 [FD] The first difference, FD, estimator for the FE model An alternative way to eliminate the individual effects α i is to take first differences (wrt time) of the FE model. or y it y i,t 1 = (x it x i,t 1 ) β + (u it u i,t 1 ) y it = x it β + u it Here also, all variables, which are only indiv specific - they do not change with t - drop out. Estimation with OLS gives the first-difference estimator ˆβ FD = ( i T t=2 x it x it ) 1 i T x it y it t=2
28 28 / 63 [FD] Properties of the FD estimator Consistency for N : if E[ x it u it ] = 0. This is slightly less demanding than for the within estimator, which is based on strict exogeneity. FD allows e.g. correlation between x it and u i,t 2. The FD estimator is slightly less efficient than the FE, as u it exhibits serial correlation, even if u it are uncorrelated. FD looses one time dimension for each i. FE looses one degree of freedom for each i by using ȳ i, x i.
29 The static linear model estimation of the random effects model 29 / 63
30 30 / 63 Estimation of the random effects model y it = β 0 + x it β + α i + u it, u it iid(0, σ 2 u), α i iid(0, σ 2 α) where (α i + u it ) is an error consisting of 2 components: an individual specific component, which does not vary over time, α i. a remainder, which is uncorrelated wrt i and t, u it. α i and u it are mutually independent, and indep of all x js. As simple OLS does not take this special error structure into account, so GLS is used.
31 31 / 63 [RE] GLS estimation In the following we stack the errors of individual i, α i and u it. Ie, we put each of them into a column vector. (α i + u it ) reads then as α i ι T + u i α i is constant for individual i ι T = (1,..., 1) is a (T 1) vector of only ones. u i = (u i1,..., u it ) is a vector collecting all u it s for indiv i. I T is the T -dimensional identity matrix. Since the errors of different indiv s are independent, the covar-matrix consists of N identical blocks in the diagonal. Block i is V(α i ι T + u i ) = Ω = σ 2 α ι T ι T + σ2 u I T Remark: The inverse of a block-diag matrix is a block-diag matrix with the inverse blocks in the diag. So it is enough to consider indiv blocks.
32 32 / 63 [RE] GLS estimation GLS corresponds to premultiplying the vectors (y i1,..., y it ), etc. by Ω 1/2. Where [ ] Ω 1 = σu 2 I T ψ ι T ι T σ 2 α with ψ = σu 2 + T σα 2 and ψ = σ 2 u σ 2 u + T σ 2 α Remark: The GLS for the classical regression model is where u.(0, Ω) is heteroscedastic. ˆβ GLS = (X Ω 1 X) 1 X Ω 1 y
33 33 / 63 [RE] GLS estimator The GLS estimator can be written as ( ˆβ GLS = (x it x i )(x it x i ) + ψ i t i ( (x it x i )(y it ȳ i ) + ψ i t i T ( x i x)( x i x) ) 1 T ( x i x)(ȳ i ȳ) ) If ψ = 0, (σ 2 u = 0), the GLS estimator is equal to the FE estimator. As if N individual intercepts were included. If ψ = 1, (σ 2 α = 0). The GLS becomes the OLS estimator. Only 1 overall intercept is included, as in the pooled model. If T then ψ 0. I.e.: The FE and RE estimators for β are equivalent for large T. (But not for T fix and N.) Remark: ( i t + i T )/(N T ) is finite as N or T or both.
34 34 / 63 Interpretation of within and between, ψ = 1 Within refers to the variation within one individual (over time), between measures the variation between the individuals. The within and between components (y it ȳ) = (y it ȳ i ) + (ȳ i ȳ) are orthogonal. So (y it ȳ) 2 = i t i (y it ȳ i ) 2 + T t i (ȳ i ȳ) 2 The variance of y may be decomposed into the sum of variance within and the variance between. ȳ i = (1/T ) t y it and ȳ = (1/(N T ) i t y it
35 35 / 63 [RE] Properties of the GLS GLS is unbiased, if the x s are independent of all u it and α i. The GLS will be more efficient than OLS in general under the RE assumptions. Consistency for N (T fix, or T ): if E[(x it x i )u it ] = 0 and E[ x i α i ] = 0 holds. Under weak conditions (errors need not be normal) the feasible GLS is asymptotically normal.
36 Comparison and testing of FE and RE 36 / 63
37 37 / 63 Interpretation: FE and RE Fixed effects: The distribution of y it is seen conditional on x it and individual dummies d i. E[y it x it, d i ] = x it β + α i This is plausible if the individuals are not a random draw, like in samples of large companies, countries, sectors. Random effects: The distribution of y it is not conditional on single individual characteristics. Arbitrary indiv effects have a fixed variance. Conclusions wrt a population are drawn. E[y it x it, 1] = x it β + β 0 If the d i are expected to be correlated with the x s, FE is preferred. RE increases only efficiency, if exogeneity is guaranteed.
38 38 / 63 Interpretation: FE and RE (T fix) The FE estimator, ˆβ FE, is consistent (N ) and efficient under the FE model assumptions. consistent, but not efficient, under the RE model assumptions, as the correlation structure of the errors is not taken into account (replaced only by different intercepts). As the error variance is not estimated consistently, the t-values are not correct. The RE estimator is consistent (N ) and efficient under the assumptions of the RE model. not consistent under the FE assumptions, as the true explanatory vars (d i, x it ) are correlated with (α i + u it ) [with ˆψ 0 and x i x biased for fix T ].
39 39 / 63 Hausman test Hausman tests the H 0 that x it and α i are uncorrelated. We compare therefore two estimators one, that is consistent under both hypotheses, and one, that is consistent (and efficient) only under the null. A significant difference between both indicates that H 0 is unlikely to hold. H 0 is the RE model H A is the FE model y it = β 0 + x it β + α i + u it y it = α i + x it β + u it ˆβ RE is consistent (and efficient), under H 0, not under H A. ˆβFE is consistent, under H 0 and H A.
40 40 / 63 Test FE against RE: Hausman test We consider the difference of both estimators. If it is large, we reject the null. Since ˆβ RE is efficient under H 0, it holds The Hausman statistic under H 0 is ˆβ FE ˆβ RE V( ˆβ FE ˆβ RE ) = V( ˆβ FE ) V( ˆβ RE ) ( ˆβ FE ˆβ RE ) [ˆV( ˆβ FE ) ˆV( ˆβ RE )] 1 ( ˆβ FE ˆβ RE ) asy χ 2 (K ) If in finite samples [ˆV( ˆβ FE ) ˆV( ˆβ RE )] is not positive definite, testing is performed only for a subset of β.
41 41 / 63 Comparison via goodness-of-fit, R 2 wrt ˆβ We are often interested in a distinction between the within R 2 and the between R 2. The within R 2 for any arbitrary estimator is given by R 2 within ( ˆβ) = Corr 2 [ŷ it ŷ i, y it ȳ i ] and between R 2 by R 2 between ( ˆβ) = Corr 2 [ŷ i, ȳ i ] where ŷ it = x it ˆβ, ŷ i = x i ˆβ and ŷ it ŷ i = (x it x i ) ˆβ. The overall R 2 is R 2 overall ( ˆβ) = Corr 2 [ŷ it, y it ]
42 42 / 63 Goodness-of-fit, R 2 The usual R 2 is valid for comparison of the pooled model estimated by OLS and the FE model. The comparison of the FE and RE via R 2 is not valid as in the FE the α i s are considered as explanatory variables, while in the RE model they belong to the unexplained error. Comparison via R 2 should be done only within the same class of models and estimators.
43 43 / 63 Testing for autocorrelation A generalization of the Durbin-Watson statistic for autocorr of order 1 can be formulated T i t=2 DW p = (û it û i,t 1 ) 2 i t û2 it The critical values depend on T, N and K. For T [6, 10] and K [3, 9] the following approximate lower and upper bounds can be used. N = 100 N = 500 N = 1000 d L d U d L d U d L d U
44 44 / 63 Testing for heteroscedasticity A generalization of the Breusch-Pagan test is applicable. σ 2 u is tested whether it depends on a set of J third variables z. V(u it ) = σ 2 h(z it γ) where for the function h(.), h(0) = 1 and h(.) > 0 holds. The null hypothesis is γ = 0. The N(T 1) multiple of R 2 u of the auxiliary regression û 2 it = σ 2 h(z it γ) + v it is distributed under the H 0 asymptotically χ 2 (J) with J degrees of freedom. N(T 1)R 2 u a χ 2 (J)
45 Dynamic panel data models 45 / 63
46 46 / 63 Dynamic panel data models The dynamic model with one lagged dependent without exogenous variables, γ < 1, is y it = γ y i,t 1 + α i + u it, u it iid(0, σ 2 u)
47 47 / 63 Endogeneity problem Here, y i,t 1 depends positively on α i : This is simple to see, when inspecting the model for period (t 1) y i,t 1 = γ y i,t 2 + α i + u i,t 1 There is an endogeneity problem. OLS or GLS will be inconsistent for N and T fixed, both for FE and RE. (Easy to see for RE.) The finite sample bias can be substantial for small T. E.g. if γ = 0.5, T = 10, and N plim N ˆγ FE = 0.33 However, as in the univariate model, if in addition T, we obtain a consistent estimator. But T is i.g. small for panel data.
48 48 / 63 The first difference estimator (FD) Using the 1st difference estimator (FD), which eliminates the α i s y it = γ y i,t 1 + u it y it y i,t 1 = γ(y i,t 1 y i,t 2 ) + (u it u i,t 1 ) is no help, since y i,t 1 and u i,t 1 are correlated even when T. We stay with the FD model, as the exogeneity requirements are less restrictive, and use an IV estimator. For that purpose we look also at y i,t 1 y i,t 1 y i,t 2 = γ(y i,t 2 y i,t 3 ) + (u i,t 1 u i,t 2 )
49 49 / 63 [FD] IV estimation, Anderson-Hsiao Instrumental variable estimators, IV, have been proposed by Anderson-Hsiao, as they are consistent with N and finite T. Choice of the instruments for (y i,t 1 y i,t 2 ): Instrument y i,t 2 as proxy is correlated with (y i,t 1 y i,t 2 ), but not with u i,t 1 or u i,t, and so u i,t. Instrument (y i,t 2 y i,t 3 ) as proxy for (y i,t 1 y i,t 2 ) sacrifies one more sample period.
50 50 / 63 [FD] GMM estimation, Arellano-Bond The Arellano-Bond (also Arellano-Bover) method of moments estimator is consistent. The moment conditions use the properties of the instruments y i,t j, j 2 to be uncorrelated with the future errors u it and u i,t 1. We obtain an increasing number of moment conditions for t = 3, 4,..., T. t = 3 : E[(u i,3 u i,2 )y i,1 ] = 0 t = 4 : E[(u i,4 u i,3 )y i,2 ] = 0, E[(u i,4 u i,3 )y i,1 ] = 0 t = 5 : E[(u i,5 u i,4 )y i,3 ] = 0,..., E[(u i,5 u i,4 )y i,1 ] =
51 51 / 63 [FD] GMM estimation, Arellano-Bond We define the (T 2) 1 vector u i = [(u i,3 u i,2 ),..., (u i,t u i,t 1 )] and a (T 2) (T 2) matrix of instruments y i,1 y i,1... y i,1 Z i 0 y = i,2... y i, y i,t 2
52 [FD] GMM estimation, Arellano-Bond, without x s Ignoring exogenous variables, for y it = γ y i,t 1 + u it, E[Z i u i ] = E[Z i ( y i γ y i, 1 )] = 0 The number of moment conditions are (T 2). This number exceeds i.g. the number of unknown coefficients, so γ is estimated by minimizing the quadratic expression [ 1 min γ N i ] [ 1 Z i ( y i γ y i, 1 ) W N N i ] Z i ( y i γ y i, 1 ) with a weighting matrix W N. The optimal matrix W N yielding an asy efficient estimator is the inverse of the covariance matrix of the sample moments. 52 / 63
53 53 / 63 [FD] GMM estimation, Arellano-Bond, without x s The W N matrix can be estimated directly from the data after a first consistent estimation step. Under weak regularity conditions the GMM estimator is asy normal for N for fixed T, T > 2 using our instruments. It is also consistent for N and T, though the number of moment conditions as T. It is advisable to limit the number of moment conditions.
54 54 / 63 [FD] GMM estimation, Arellano-Bond, with x s The dynamic panel data model with exogenous variables is y it = x it β + γ y i,t 1 + α i + u it, u it iid(0, σ 2 u) As also exogenous x s are included in the model additional moment conditions can be formulated: For strictly exogenous vars, E[x is u it ] = 0 for all s, t, E[x is u it ] = 0 For predetermined (not strictly exogenous) vars, E[x is u it ] = 0 for s t E[x i,t j u it ] = 0 j = 1,..., t 1
55 55 / 63 GMM estimation, Arellano-Bond, with x s So there are a lot of possible moment restrictions both for differences as well as for levels, and so a variety of GMM estimators. GMM estimation may be combined with both FE and RE. Here also, the RE estimator is identical to the FE estimator with T.
56 Further topics 56 / 63
57 57 / 63 Further topics We augment the FE model with dummy variables indicating both individuals and time periods. y it = δ i d i + δ t d t + x it β + u it Common unit roots can be tested and modeled. which is transformed to y it = α i + γ i y i,t 1 + u it and tested wrt H 0 : π i = 0 for all i y it = α i + π i y i,t 1 + u it
58 58 / 63 Panel data models: further topics Cointegration tests. Limited dependent models, like binary choice, logit, probit models, etc.
59 Exercises and References 59 / 63
60 60 / 63 Exercises Choose one out of (1, 2), and 3 out of (3G, 4G, 5G, 6G). 7G is compulsory. 1 The fixed effects model with group and time effects is y it = α i d i + δ t d t + x it β + u it where d i are dummies for individuals, and d t dummies for time periods. Specify the model correctly when an overall intercept is included. 2 Show that holds. (y it ȳ) 2 = t i i (y it ȳ i ) 2 + T t i (ȳ i ȳ) 2
61 61 / 63 Exercises 3G Choose a subset of firms for the Grunfeld data and compare the pooling and the fixed effects model. Use Ex5_panel_data_R.txt. 4G Choose a subset of firms for the Grunfeld data and compare the fixed effects and the random effects model. Use Ex5_panel_data_R.txt. 5G Choose a subset of firms for the Grunfeld/EmplUK data and estimate a suitable dynamic panel data model. Use Ex5_panel_data_R.txt. 6G Compute the Durbin-Watson statistic for panel data in R. Test for serial correlation in the fixed effects model of a data set of your choice.
62 62 / 63 Exercises 7G (Compulsory) Prepare a short summary about the Illustration 10.5: Explaining capital structure, in Verbeek, p , covering the following issues: the empirical problem/question posed the model(s) under consideration/ the relations between the variables suggested by economic theory the formal model the data (source, coverage, proxies for the variables in the theoretical model) the estimation method(s), empirical specification of the model, comparison of models results/conclusions Delivery of an electronic version of 7G not later than 1 week after the final test.
63 63 / 63 References Verbeek 10
What s New in Econometrics? Lecture 8 Cluster and Stratified Sampling
What s New in Econometrics? Lecture 8 Cluster and Stratified Sampling Jeff Wooldridge NBER Summer Institute, 2007 1. The Linear Model with Cluster Effects 2. Estimation with a Small Number of Groups and
More informationSYSTEMS OF REGRESSION EQUATIONS
SYSTEMS OF REGRESSION EQUATIONS 1. MULTIPLE EQUATIONS y nt = x nt n + u nt, n = 1,...,N, t = 1,...,T, x nt is 1 k, and n is k 1. This is a version of the standard regression model where the observations
More informationChapter 10: Basic Linear Unobserved Effects Panel Data. Models:
Chapter 10: Basic Linear Unobserved Effects Panel Data Models: Microeconomic Econometrics I Spring 2010 10.1 Motivation: The Omitted Variables Problem We are interested in the partial effects of the observable
More informationChapter 2. Dynamic panel data models
Chapter 2. Dynamic panel data models Master of Science in Economics - University of Geneva Christophe Hurlin, Université d Orléans Université d Orléans April 2010 Introduction De nition We now consider
More informationChapter 4: Vector Autoregressive Models
Chapter 4: Vector Autoregressive Models 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie IV.1 Vector Autoregressive Models (VAR)...
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More information2. Linear regression with multiple regressors
2. Linear regression with multiple regressors Aim of this section: Introduction of the multiple regression model OLS estimation in multiple regression Measures-of-fit in multiple regression Assumptions
More informationClustering in the Linear Model
Short Guides to Microeconometrics Fall 2014 Kurt Schmidheiny Universität Basel Clustering in the Linear Model 2 1 Introduction Clustering in the Linear Model This handout extends the handout on The Multiple
More informationChapter 6: Multivariate Cointegration Analysis
Chapter 6: Multivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie VI. Multivariate Cointegration
More information1 Teaching notes on GMM 1.
Bent E. Sørensen January 23, 2007 1 Teaching notes on GMM 1. Generalized Method of Moment (GMM) estimation is one of two developments in econometrics in the 80ies that revolutionized empirical work in
More informationPanel Data: Linear Models
Panel Data: Linear Models Laura Magazzini University of Verona laura.magazzini@univr.it http://dse.univr.it/magazzini Laura Magazzini (@univr.it) Panel Data: Linear Models 1 / 45 Introduction Outline What
More informationPractical. I conometrics. data collection, analysis, and application. Christiana E. Hilmer. Michael J. Hilmer San Diego State University
Practical I conometrics data collection, analysis, and application Christiana E. Hilmer Michael J. Hilmer San Diego State University Mi Table of Contents PART ONE THE BASICS 1 Chapter 1 An Introduction
More informationThe VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.
Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium
More informationEconometrics Simple Linear Regression
Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight
More informationFULLY MODIFIED OLS FOR HETEROGENEOUS COINTEGRATED PANELS
FULLY MODIFIED OLS FOR HEEROGENEOUS COINEGRAED PANELS Peter Pedroni ABSRAC his chapter uses fully modified OLS principles to develop new methods for estimating and testing hypotheses for cointegrating
More informationFIXED EFFECTS AND RELATED ESTIMATORS FOR CORRELATED RANDOM COEFFICIENT AND TREATMENT EFFECT PANEL DATA MODELS
FIXED EFFECTS AND RELATED ESTIMATORS FOR CORRELATED RANDOM COEFFICIENT AND TREATMENT EFFECT PANEL DATA MODELS Jeffrey M. Wooldridge Department of Economics Michigan State University East Lansing, MI 48824-1038
More informationEconometric Methods for Panel Data
Based on the books by Baltagi: Econometric Analysis of Panel Data and by Hsiao: Analysis of Panel Data Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies
More informationModule 5: Multiple Regression Analysis
Using Statistical Data Using to Make Statistical Decisions: Data Multiple to Make Regression Decisions Analysis Page 1 Module 5: Multiple Regression Analysis Tom Ilvento, University of Delaware, College
More informationCOURSES: 1. Short Course in Econometrics for the Practitioner (P000500) 2. Short Course in Econometric Analysis of Cointegration (P000537)
Get the latest knowledge from leading global experts. Financial Science Economics Economics Short Courses Presented by the Department of Economics, University of Pretoria WITH 2015 DATES www.ce.up.ac.za
More informationWooldridge, Introductory Econometrics, 3d ed. Chapter 12: Serial correlation and heteroskedasticity in time series regressions
Wooldridge, Introductory Econometrics, 3d ed. Chapter 12: Serial correlation and heteroskedasticity in time series regressions What will happen if we violate the assumption that the errors are not serially
More informationMultiple Linear Regression in Data Mining
Multiple Linear Regression in Data Mining Contents 2.1. A Review of Multiple Linear Regression 2.2. Illustration of the Regression Process 2.3. Subset Selection in Linear Regression 1 2 Chap. 2 Multiple
More informationAssociation Between Variables
Contents 11 Association Between Variables 767 11.1 Introduction............................ 767 11.1.1 Measure of Association................. 768 11.1.2 Chapter Summary.................... 769 11.2 Chi
More informationMULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS
MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level of Significance
More informationON THE ROBUSTNESS OF FIXED EFFECTS AND RELATED ESTIMATORS IN CORRELATED RANDOM COEFFICIENT PANEL DATA MODELS
ON THE ROBUSTNESS OF FIXED EFFECTS AND RELATED ESTIMATORS IN CORRELATED RANDOM COEFFICIENT PANEL DATA MODELS Jeffrey M. Wooldridge THE INSTITUTE FOR FISCAL STUDIES DEPARTMENT OF ECONOMICS, UCL cemmap working
More informationLOGIT AND PROBIT ANALYSIS
LOGIT AND PROBIT ANALYSIS A.K. Vasisht I.A.S.R.I., Library Avenue, New Delhi 110 012 amitvasisht@iasri.res.in In dummy regression variable models, it is assumed implicitly that the dependent variable Y
More informationMISSING DATA TECHNIQUES WITH SAS. IDRE Statistical Consulting Group
MISSING DATA TECHNIQUES WITH SAS IDRE Statistical Consulting Group ROAD MAP FOR TODAY To discuss: 1. Commonly used techniques for handling missing data, focusing on multiple imputation 2. Issues that could
More informationECON 142 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE #2
University of California, Berkeley Prof. Ken Chay Department of Economics Fall Semester, 005 ECON 14 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE # Question 1: a. Below are the scatter plots of hourly wages
More informationMarketing Mix Modelling and Big Data P. M Cain
1) Introduction Marketing Mix Modelling and Big Data P. M Cain Big data is generally defined in terms of the volume and variety of structured and unstructured information. Whereas structured data is stored
More informationFactor analysis. Angela Montanari
Factor analysis Angela Montanari 1 Introduction Factor analysis is a statistical model that allows to explain the correlations between a large number of observed correlated variables through a small number
More informationFinancial Risk Management Exam Sample Questions/Answers
Financial Risk Management Exam Sample Questions/Answers Prepared by Daniel HERLEMONT 1 2 3 4 5 6 Chapter 3 Fundamentals of Statistics FRM-99, Question 4 Random walk assumes that returns from one time period
More informationCorrelated Random Effects Panel Data Models
INTRODUCTION AND LINEAR MODELS Correlated Random Effects Panel Data Models IZA Summer School in Labor Economics May 13-19, 2013 Jeffrey M. Wooldridge Michigan State University 1. Introduction 2. The Linear
More informationIntroduction to Regression Models for Panel Data Analysis. Indiana University Workshop in Methods October 7, 2011. Professor Patricia A.
Introduction to Regression Models for Panel Data Analysis Indiana University Workshop in Methods October 7, 2011 Professor Patricia A. McManus Panel Data Analysis October 2011 What are Panel Data? Panel
More informationIntroduction to General and Generalized Linear Models
Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby
More informationThe Loss in Efficiency from Using Grouped Data to Estimate Coefficients of Group Level Variables. Kathleen M. Lang* Boston College.
The Loss in Efficiency from Using Grouped Data to Estimate Coefficients of Group Level Variables Kathleen M. Lang* Boston College and Peter Gottschalk Boston College Abstract We derive the efficiency loss
More informationEFFECT OF INVENTORY MANAGEMENT EFFICIENCY ON PROFITABILITY: CURRENT EVIDENCE FROM THE U.S. MANUFACTURING INDUSTRY
EFFECT OF INVENTORY MANAGEMENT EFFICIENCY ON PROFITABILITY: CURRENT EVIDENCE FROM THE U.S. MANUFACTURING INDUSTRY Seungjae Shin, Mississippi State University Kevin L. Ennis, Mississippi State University
More informationEconometric analysis of the Belgian car market
Econometric analysis of the Belgian car market By: Prof. dr. D. Czarnitzki/ Ms. Céline Arts Tim Verheyden Introduction In contrast to typical examples from microeconomics textbooks on homogeneous goods
More informationSimple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
More informationNote 2 to Computer class: Standard mis-specification tests
Note 2 to Computer class: Standard mis-specification tests Ragnar Nymoen September 2, 2013 1 Why mis-specification testing of econometric models? As econometricians we must relate to the fact that the
More informationFORECASTING DEPOSIT GROWTH: Forecasting BIF and SAIF Assessable and Insured Deposits
Technical Paper Series Congressional Budget Office Washington, DC FORECASTING DEPOSIT GROWTH: Forecasting BIF and SAIF Assessable and Insured Deposits Albert D. Metz Microeconomic and Financial Studies
More informationNCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )
Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates
More informationINDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition)
INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) Abstract Indirect inference is a simulation-based method for estimating the parameters of economic models. Its
More informationPanel Data Analysis in Stata
Panel Data Analysis in Stata Anton Parlow Lab session Econ710 UWM Econ Department??/??/2010 or in a S-Bahn in Berlin, you never know.. Our plan Introduction to Panel data Fixed vs. Random effects Testing
More informationIMPACT EVALUATION: INSTRUMENTAL VARIABLE METHOD
REPUBLIC OF SOUTH AFRICA GOVERNMENT-WIDE MONITORING & IMPACT EVALUATION SEMINAR IMPACT EVALUATION: INSTRUMENTAL VARIABLE METHOD SHAHID KHANDKER World Bank June 2006 ORGANIZED BY THE WORLD BANK AFRICA IMPACT
More informationPart 2: Analysis of Relationship Between Two Variables
Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable
More informationUnit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression
Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a
More informationEmployer-Provided Health Insurance and Labor Supply of Married Women
Upjohn Institute Working Papers Upjohn Research home page 2011 Employer-Provided Health Insurance and Labor Supply of Married Women Merve Cebi University of Massachusetts - Dartmouth and W.E. Upjohn Institute
More informationChapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem
Chapter Vector autoregressions We begin by taking a look at the data of macroeconomics. A way to summarize the dynamics of macroeconomic data is to make use of vector autoregressions. VAR models have become
More informationPanel Data Econometrics
Panel Data Econometrics Master of Science in Economics - University of Geneva Christophe Hurlin, Université d Orléans University of Orléans January 2010 De nition A longitudinal, or panel, data set is
More information1. THE LINEAR MODEL WITH CLUSTER EFFECTS
What s New in Econometrics? NBER, Summer 2007 Lecture 8, Tuesday, July 31st, 2.00-3.00 pm Cluster and Stratified Sampling These notes consider estimation and inference with cluster samples and samples
More informationAPPROXIMATING THE BIAS OF THE LSDV ESTIMATOR FOR DYNAMIC UNBALANCED PANEL DATA MODELS. Giovanni S.F. Bruno EEA 2004-1
ISTITUTO DI ECONOMIA POLITICA Studi e quaderni APPROXIMATING THE BIAS OF THE LSDV ESTIMATOR FOR DYNAMIC UNBALANCED PANEL DATA MODELS Giovanni S.F. Bruno EEA 2004-1 Serie di econometria ed economia applicata
More informationIntroduction to Regression and Data Analysis
Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it
More informationI n d i a n a U n i v e r s i t y U n i v e r s i t y I n f o r m a t i o n T e c h n o l o g y S e r v i c e s
I n d i a n a U n i v e r s i t y U n i v e r s i t y I n f o r m a t i o n T e c h n o l o g y S e r v i c e s Linear Regression Models for Panel Data Using SAS, Stata, LIMDEP, and SPSS * Hun Myoung Park,
More informationThe Effect of R&D Expenditures on Stock Returns, Price and Volatility
Master Degree Project in Finance The Effect of R&D Expenditures on Stock Returns, Price and Volatility A study on biotechnological and pharmaceutical industry in the US market Aleksandra Titi Supervisor:
More informationA Subset-Continuous-Updating Transformation on GMM Estimators for Dynamic Panel Data Models
Article A Subset-Continuous-Updating Transformation on GMM Estimators for Dynamic Panel Data Models Richard A. Ashley 1, and Xiaojin Sun 2,, 1 Department of Economics, Virginia Tech, Blacksburg, VA 24060;
More informationLecture 3: Differences-in-Differences
Lecture 3: Differences-in-Differences Fabian Waldinger Waldinger () 1 / 55 Topics Covered in Lecture 1 Review of fixed effects regression models. 2 Differences-in-Differences Basics: Card & Krueger (1994).
More informationStatistics 2014 Scoring Guidelines
AP Statistics 2014 Scoring Guidelines College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home
More informationproblem arises when only a non-random sample is available differs from censored regression model in that x i is also unobserved
4 Data Issues 4.1 Truncated Regression population model y i = x i β + ε i, ε i N(0, σ 2 ) given a random sample, {y i, x i } N i=1, then OLS is consistent and efficient problem arises when only a non-random
More informationFrom the help desk: Swamy s random-coefficients model
The Stata Journal (2003) 3, Number 3, pp. 302 308 From the help desk: Swamy s random-coefficients model Brian P. Poi Stata Corporation Abstract. This article discusses the Swamy (1970) random-coefficients
More informationDepartment of Economics
Department of Economics On Testing for Diagonality of Large Dimensional Covariance Matrices George Kapetanios Working Paper No. 526 October 2004 ISSN 1473-0278 On Testing for Diagonality of Large Dimensional
More information4. Simple regression. QBUS6840 Predictive Analytics. https://www.otexts.org/fpp/4
4. Simple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/4 Outline The simple linear model Least squares estimation Forecasting with regression Non-linear functional forms Regression
More informationIdentifying Non-linearities In Fixed Effects Models
Identifying Non-linearities In Fixed Effects Models Craig T. McIntosh and Wolfram Schlenker October 2006 Abstract We discuss the use of quadratic terms in models which include fixed effects or dummy variables.
More informationSimple Regression Theory II 2010 Samuel L. Baker
SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the
More informationTesting for Granger causality between stock prices and economic growth
MPRA Munich Personal RePEc Archive Testing for Granger causality between stock prices and economic growth Pasquale Foresti 2006 Online at http://mpra.ub.uni-muenchen.de/2962/ MPRA Paper No. 2962, posted
More informationAn Investigation of the Statistical Modelling Approaches for MelC
An Investigation of the Statistical Modelling Approaches for MelC Literature review and recommendations By Jessica Thomas, 30 May 2011 Contents 1. Overview... 1 2. The LDV... 2 2.1 LDV Specifically in
More information3.1 Least squares in matrix form
118 3 Multiple Regression 3.1 Least squares in matrix form E Uses Appendix A.2 A.4, A.6, A.7. 3.1.1 Introduction More than one explanatory variable In the foregoing chapter we considered the simple regression
More informationState Space Time Series Analysis
State Space Time Series Analysis p. 1 State Space Time Series Analysis Siem Jan Koopman http://staff.feweb.vu.nl/koopman Department of Econometrics VU University Amsterdam Tinbergen Institute 2011 State
More informationPlease follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software
STATA Tutorial Professor Erdinç Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software 1.Wald Test Wald Test is used
More informationLongitudinal Meta-analysis
Quality & Quantity 38: 381 389, 2004. 2004 Kluwer Academic Publishers. Printed in the Netherlands. 381 Longitudinal Meta-analysis CORA J. M. MAAS, JOOP J. HOX and GERTY J. L. M. LENSVELT-MULDERS Department
More informationSTATISTICA Formula Guide: Logistic Regression. Table of Contents
: Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary
More informationReview Jeopardy. Blue vs. Orange. Review Jeopardy
Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?
More informationFinancial TIme Series Analysis: Part II
Department of Mathematics and Statistics, University of Vaasa, Finland January 29 February 13, 2015 Feb 14, 2015 1 Univariate linear stochastic models: further topics Unobserved component model Signal
More informationRegression Analysis (Spring, 2000)
Regression Analysis (Spring, 2000) By Wonjae Purposes: a. Explaining the relationship between Y and X variables with a model (Explain a variable Y in terms of Xs) b. Estimating and testing the intensity
More informationChapter 3: The Multiple Linear Regression Model
Chapter 3: The Multiple Linear Regression Model Advanced Econometrics - HEC Lausanne Christophe Hurlin University of Orléans November 23, 2013 Christophe Hurlin (University of Orléans) Advanced Econometrics
More informationMultiple Regression: What Is It?
Multiple Regression Multiple Regression: What Is It? Multiple regression is a collection of techniques in which there are multiple predictors of varying kinds and a single outcome We are interested in
More informationLasso on Categorical Data
Lasso on Categorical Data Yunjin Choi, Rina Park, Michael Seo December 14, 2012 1 Introduction In social science studies, the variables of interest are often categorical, such as race, gender, and nationality.
More informationSolución del Examen Tipo: 1
Solución del Examen Tipo: 1 Universidad Carlos III de Madrid ECONOMETRICS Academic year 2009/10 FINAL EXAM May 17, 2010 DURATION: 2 HOURS 1. Assume that model (III) verifies the assumptions of the classical
More informationMODELS FOR PANEL DATA Q
Greene-2140242 book November 23, 2010 12:28 11 MODELS FOR PANEL DATA Q 11.1 INTRODUCTION Data sets that combine time series and cross sections are common in economics. The published statistics of the OECD
More informationExample: Boats and Manatees
Figure 9-6 Example: Boats and Manatees Slide 1 Given the sample data in Table 9-1, find the value of the linear correlation coefficient r, then refer to Table A-6 to determine whether there is a significant
More informationVector Time Series Model Representations and Analysis with XploRe
0-1 Vector Time Series Model Representations and Analysis with plore Julius Mungo CASE - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin mungo@wiwi.hu-berlin.de plore MulTi Motivation
More informationExact Nonparametric Tests for Comparing Means - A Personal Summary
Exact Nonparametric Tests for Comparing Means - A Personal Summary Karl H. Schlag European University Institute 1 December 14, 2006 1 Economics Department, European University Institute. Via della Piazzuola
More informationModule 3: Correlation and Covariance
Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis
More informationPremium Copayments and the Trade-off between Wages and Employer-Provided Health Insurance. February 2011
PRELIMINARY DRAFT DO NOT CITE OR CIRCULATE COMMENTS WELCOMED Premium Copayments and the Trade-off between Wages and Employer-Provided Health Insurance February 2011 By Darren Lubotsky School of Labor &
More informationCausal Forecasting Models
CTL.SC1x -Supply Chain & Logistics Fundamentals Causal Forecasting Models MIT Center for Transportation & Logistics Causal Models Used when demand is correlated with some known and measurable environmental
More informationIntroduction to Quantitative Methods
Introduction to Quantitative Methods October 15, 2009 Contents 1 Definition of Key Terms 2 2 Descriptive Statistics 3 2.1 Frequency Tables......................... 4 2.2 Measures of Central Tendencies.................
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationMultivariate normal distribution and testing for means (see MKB Ch 3)
Multivariate normal distribution and testing for means (see MKB Ch 3) Where are we going? 2 One-sample t-test (univariate).................................................. 3 Two-sample t-test (univariate).................................................
More informationAccounting for Time-Varying Unobserved Ability Heterogeneity within Education Production Functions
Accounting for Time-Varying Unobserved Ability Heterogeneity within Education Production Functions Weili Ding Queen s University Steven F. Lehrer Queen s University and NBER July 2008 Abstract Traditional
More informationPOLYNOMIAL AND MULTIPLE REGRESSION. Polynomial regression used to fit nonlinear (e.g. curvilinear) data into a least squares linear regression model.
Polynomial Regression POLYNOMIAL AND MULTIPLE REGRESSION Polynomial regression used to fit nonlinear (e.g. curvilinear) data into a least squares linear regression model. It is a form of linear regression
More informationQuadratic forms Cochran s theorem, degrees of freedom, and all that
Quadratic forms Cochran s theorem, degrees of freedom, and all that Dr. Frank Wood Frank Wood, fwood@stat.columbia.edu Linear Regression Models Lecture 1, Slide 1 Why We Care Cochran s theorem tells us
More informationGood luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:
Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours
More informationMultivariate Analysis of Health Survey Data
10 Multivariate Analysis of Health Survey Data The most basic description of health sector inequality is given by the bivariate relationship between a health variable and some indicator of socioeconomic
More informationRobust Inferences from Random Clustered Samples: Applications Using Data from the Panel Survey of Income Dynamics
Robust Inferences from Random Clustered Samples: Applications Using Data from the Panel Survey of Income Dynamics John Pepper Assistant Professor Department of Economics University of Virginia 114 Rouss
More informationTesting The Quantity Theory of Money in Greece: A Note
ERC Working Paper in Economic 03/10 November 2003 Testing The Quantity Theory of Money in Greece: A Note Erdal Özmen Department of Economics Middle East Technical University Ankara 06531, Turkey ozmen@metu.edu.tr
More informationOnline Appendices to the Corporate Propensity to Save
Online Appendices to the Corporate Propensity to Save Appendix A: Monte Carlo Experiments In order to allay skepticism of empirical results that have been produced by unusual estimators on fairly small
More information1 Introduction. 2 The Econometric Model. Panel Data: Fixed and Random Effects. Short Guides to Microeconometrics Fall 2015
Short Guides to Microeconometrics Fall 2015 Kurt Schmidheiny Unversität Basel Panel Data: Fixed and Random Effects 2 Panel Data: Fixed and Random Effects 1 Introduction In panel data, individuals (persons,
More informationMinimum LM Unit Root Test with One Structural Break. Junsoo Lee Department of Economics University of Alabama
Minimum LM Unit Root Test with One Structural Break Junsoo Lee Department of Economics University of Alabama Mark C. Strazicich Department of Economics Appalachian State University December 16, 2004 Abstract
More informationPITFALLS IN TIME SERIES ANALYSIS. Cliff Hurvich Stern School, NYU
PITFALLS IN TIME SERIES ANALYSIS Cliff Hurvich Stern School, NYU The t -Test If x 1,..., x n are independent and identically distributed with mean 0, and n is not too small, then t = x 0 s n has a standard
More informationLinear Models for Continuous Data
Chapter 2 Linear Models for Continuous Data The starting point in our exploration of statistical models in social research will be the classical linear model. Stops along the way include multiple linear
More informationOverview of Factor Analysis
Overview of Factor Analysis Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone: (205) 348-4431 Fax: (205) 348-8648 August 1,
More informationSpatial panel models
Spatial panel models J Paul Elhorst University of Groningen, Department of Economics, Econometrics and Finance PO Box 800, 9700 AV Groningen, the Netherlands Phone: +31 50 3633893, Fax: +31 50 3637337,
More information