Appunti quantitative methods in financial economics

AUTOREGRESSIVE DISTRIBUTED LAG (ARDL)

Capturing the dynamic characteristics of time-series by specifying a model with a lagged dependent variable as one of the explanatory variables: Y = f(Y_t-1, X_t) or Y = f(Y_t-1, X_t, X_t-1, X_t-2)

AUTOCORRELATED ERRORS model the continuing impact of change over several periods via the error term: Y = f(X_t) + e with e = g(e_t-1)

The assumption cov(e_t, e_t-1) = cov(y_t, y_t-1) = 0 is violated. The current error e affects not just the current value of the dependent variable Y, but also its future value Y_t+1, Y_t+2,...

FINITE DISTRIBUTED LAGS MODEL

Consider a linear model in which, after q time periods (q = lags of X), changes in X no longer have an impact on Y: α + β₁X_t + β₂X_t-1 + ... + β_sX_t-s + e_t = 0

α is introduced to denote the intercept. Instead of having a number of explanatory variables, we have a number of different lags of the same variable X.

explanatory variable (ADL(s), q=s).MULTIPLIER:- TEMPORARY CHANGE: the effect of a one-unit change in X is distributed over the current and next q periods. The coefficient ß is called a DISTRIBUTED-LAG WEIGHT or S-PERIOD DELAY MULTIPLIER. The coefficient ß (s=0) is called IMPACT0MULTIPLIER.- PERMANET CHANGE: what happens if X is increased by one unit and then maintained at its new level in subsequent periods t+1, t+2,…? The immediate impact will again be ß; the total effect in period t+1 will be ß + ß , in period t+2 will be ß + ß + ß and so on 0 1 0 1 2(INTERIM MULTIPLIER). The TOTAL MULTIPLIER is the final effect on Y of the sustained increase after q or more periods have elapsed, it is given by ∑ß .sASSUMPTION:To measure a dynamic causal effect, the assumptions of stationarity (mean and variance are constant) and exogeneity must hold.α + … + ß1. Y = + ß X + ß X X + et 0 t 1 t-1 s t–s t2.

: = 00 ρe λb + b X + e^ = ß + ß X + +1 2 t t 1 2 t t–1 t– – ρe λe^ = (ß b ) +( ß b )X + +t 1 1 2 2 t t–1 t2LM = T RMODEL WITH AUTOREGRESSIVE ERROR

Our model with an AR(1) error is: ρe λY = ß + ß X + +t 1 2 t t–1 t

with: –1 ≤ ρ ≤ 1 E(λ σ 2) = 0 var(λ ) = cov(λ ,λ ) = 0λt t t s

For the previous period, the error term is: – –e = Y ß ß Xt–1 t–1 1 2 t–1ρ

Multiplying by and substituting, we get:– ρ) ρY – ρß λY = ß (1 + ß X + X +t 1 2 t t–1 2 t–1 t

The original model with the autocorrelated error term e has been transformed into a new model thattλhas an error term that is uncorrelated over time . It is possible to proceed to find estimates for ß ,t 1ρ that minimize the sum of squares of uncorrelated errors

∑λß , obtaining an estimator that is best2 tand whose standard errors are correct.

ARDL(p, q)

Including appropriate lags of Y and X without the AR(1) error restriction is preferred because by including a sufficient number of lags of Y and X we can eliminate serial correlation in the errors.

δ θ + … + θ δ δ δ + … + δ λY = + Y Y + X + X + X X +t 1 t–1 p t–p 0 t 1 t–1 2 t–2 q t–q t with p lags of Y and q lags of X.

It is a general model because can be transformed into one with only lagged X’s which go back into the infinite past:

INFINITE DISTIBUTED LAGS MODEL

α + … + eY = + ß X + ß Xt 0 t 1 t-1 t16 17

PANEL DATA MODEL

units (people, households,…) who are A panel of data consists of a group of cross-sectional observed over time. Panel data are doubly indexed by individual and time:), i = 1, …, N t = 1, …, T(X , Yit it There could be an amount

of LONG and SHORT time and NARROW and WIDE cross-sectionalunit. →UNBALANCED PANEL the number of time series observations is different across individuals.→BALANCED PANEL each individual has the same number of observations.

POOLED MODELThe data on different individuals are simply pooled together with no provision for individualdifferences that might lead to different coefficients (FULL HOMOGENEITY).αY = ß + ß X + ß X + W + eit 1 2 2it 3 3it 1 1i itW is a variable that is time invariant; the coefficients ß , ß , ß without i or t subscripts because are1i 1 2 3constant for all individuals in all time periods, and don’t allow for possible individualheterogeneity.

HYPOTHESIS:E(e ) = 0it σit2 2var(e ) = E(e ) = (homoskedasticity)itcov(e , e ) = E(e , e ) = 0it js it jscov(e , X ) = cov(e , X ) = 0it 2it it 3itBut if there are unobservable individual characteristics that are not included in the set ofexplanatory variables (but are

included in the error term) then those characteristics will lead to similar effects in different years for the same individual: ψcov(e , e ) = it is ts

When t=s then the error variance can be different in different time periods, but is constant over individuals: ψcov(e , e ) = var(e ) = it it it tt

Panel-Robust standard errors or Cluster-Robust standard errors when there are heteroskedasticity and correlation because the time-series observations on individuals are clusters; similar ideas to what we saw for heteroskedasticity and autocorrelated consistent errors (HAC).

FIXED EFFECT MODEL

We can extend the previous model to relax the assumption that all individuals have the same coefficients adding an i at the ß (FULL HETEROGENEITY):

s Y = ß + ß X + ß X + eit 1i 2i 2it 3i 3it it

A popular simplification is one where the intercepts are different for different individuals but the slope coefficients are assumed to be constant for all individuals:

Y = ß + ß X +

ß X + eit 1i 2 2it 3 3it itThe individual effects vary across individuals but are constant over time; they capture thespecificities of individual behaviors (INDIVIDUAL HETEROGENEITY); all behavioraldifferences between individuals are assumed to be captured by the intercepts that are called FIXEDEFFECTS.

We consider 2 methods for estimating the model: →

1. LEAST SQUARES DUMMY VARIABLE ESTIMATOR to include an intercept dummyvariable (indicator variable) for each individual (ex. D =1 if i=1 or 0 otherwise)1i+ … +Y = ß D + ß D ß D + ß X + ß X + eit 1,1 1i 1,2 2i 1,10 10i 2 2it 3 3it it

If the error terms e are uncorrelated with mean zero and constant variance for all observations, theitBLUE is the least squares estimator (the least squares dummy variable estimator).18 →

2. FIXED EFFECT ESTIMATOR if we have a very large number of individuals, the firsttechnique is not feasible; average the data across time, by summing both sides of the

equation and dividing by T: (1/T)∑(Y = ß + ß X + ß X + e )it 1i 2 2it 3 3it it parameters don’t change over time, we can simplify this as:

Using the fact that the ˆ = ß ˆ + ß ˆ ˆY + ß X X + ei 1i 2 2i 3 3i i

Subtracting the second equation from the first we obtain:– ˆ = – ˆ) – ˆ) – ˆ)Y Y ß (X X + ß (X X + (e eit i 2 2it 2i 3 3it 3i it i~it ~2it ~3it ~it

Y = ß X + ß X + e2 3

If one of the variables is time-invariant for each individual it is constant and the corresponding deviation from means variable would consist completely of zeros, and cannot be included.

RANDOM EFFECT MODEL

We assume that all individual differences are captured by the intercept parameters but we also recognize that the individuals in our sample were randomly selected, and thus we treat the individual differences as random rather than fixed.

Random individual differences can be included in our

model by specifying the intercept parametersto consist of a fixed part that represents the population average and random individual differencesfrom the population average: ßˆß = + u1i 1 i߈ fixed part that represents the population average; the random individual differences u are called1 iRANDOM EFFECTS and have: σ 2uE(u ) = 0 cov(u ; u ) = 0 var(u ) =i i j iTHE MODEL: Y = ß + ß X + ß X + eit 1i 2 2it 3 3it it= (߈ + u ) + ß X + ß X + e1 i 2 2it 3 3it it= ߈ + ß X + ß X + (e + u )1 2 2it 3 3it it i= ߈ λ+ ß X + ß X +1 2 2it 3 3it it→λ COMBINED ERROR composed of a component u that represents a random individual effectit i(represents unobserved heterogeneity, summarizing the unobserved factors leading to individualdifferences) and the component e which is the usual regression error (called idiosyncratic error).itERROR TERM ASSUMPTIONS:E(λ ) =