ECOM027 Research Methods Methods and tools for Empirical - PowerPoint PPT Presentation

ECOM027 – Research Methods Methods and tools for Empirical Macroeconomics Luca Rondina February 25, 2020 University of Surrey

Introduction The goal of applied macroeconomics is to design models useful for policy analysis and forecasting. The main methods are: • Time series econometrics, e.g. Vector Autoregressions (VAR) • Dynamic stochastic general equilibrium (DSGE) models, e.g. Real Business Cycles (RBC) and New Keynesian (NK) models Today: • Vector Autoregression (VAR) and Structural Vector Autoregression (SVAR) models • VAR toolbox in MATLAB 1

Introduction The goal of applied macroeconomics is to design models useful for policy analysis and forecasting. The main methods are: • Time series econometrics, e.g. Vector Autoregressions (VAR) • Dynamic stochastic general equilibrium (DSGE) models, e.g. Real Business Cycles (RBC) and New Keynesian (NK) models Today: • Vector Autoregression (VAR) and Structural Vector Autoregression (SVAR) models • VAR toolbox in MATLAB 1

Trend and cycles

Trend and cycles Most macroeconomic time series exhibit a trend. For instance, Gross Domestic Product (GDP): Figure 1: Real GDP per capita. 2 150 Billions 2009 $ 100 50 1960 1970 1980 1990 2000 2010

Trend and cycles In applied macro, we are often interested in cyclical fmuctuations of GDP. To obtain the cyclical component of the time series we have two options: • Estimate and remove the time trend. • Filter out the lower frequency components. 3

Log-linear trend We assume that the variable of interest grows linearly with time: Steps: 2. Remove the trend component to obtain the cycle. 4 y t = c + δ t + ε t 1. Estimate the δ coeffjcient to get the trend.

Log-linear trend Figure 2: Log real GDP per capita with linear trend. 5 1. Estimate the δ coeffjcient to get the trend: 500 450 400 1960 1970 1980 1990 2000 2010

Log-linear trend 2. Remove the trend component to obtain the cycle: Figure 3: % deviations of GDP per capita around linear trend. 6 10 5 % deviation 0 -5 -10 1960 1970 1980 1990 2000 2010

Filtering The time series moves at different frequencies, higher frequencies and lower frequencies. frequencies of the time series Steps: 1. Use the Hodrick–Prescott fjlter to obtain the ”slow moving” component. 2. Remove the lower frequency to get the cycle. 7 We use the Hodrick–Prescott fjlter to separate the higher and lower

Filtering 1. Use the Hodrick–Prescott fjlter to obtain the ”slow moving” component: Figure 4: Log real GDP per capita with HP fjltered trend. 8 500 450 400 350 1960 1970 1980 1990 2000 2010

Filtering 2. Remove the lower frequency to get the cycle: Figure 5: HP fjltered real GDP per capita. 9 4 2 % deviation 0 -2 -4 -6 1960 1970 1980 1990 2000 2010

Comparison: Linear trend vs HP fjlter Both approaches can successfully identify recessions: Figure 6: Linear de-trended vs HP fjltered with NBER recessions. 10

Vector Autoregression models

0 defjned by: y t t • Initial value y 0 is known • e t is an i.i.d. innovation that follows e t Vector Autoregressions (VARs) 1 Stability condition : e 2 0 (1) where: Vector Autoregressions are the multivariate equivalent of an e t 1 y t y t Recall an AR process of order one (AR1). This process is a sequence Autoregressive (AR) process. 11

Vector Autoregressions (VARs) Vector Autoregressions are the multivariate equivalent of an Autoregressive (AR) process. Recall an AR process of order one (AR1). This process is a sequence (1) where: Stability condition : 1 11 { y t } ∞ t = 0 defjned by: y t = ρ y t − 1 + e t • Initial value y 0 is known • e t is an i.i.d. innovation that follows e t ∼ N ( 0 , σ 2 e )

Vector Autoregressions (VARs) Vector Autoregressions are the multivariate equivalent of an Autoregressive (AR) process. Recall an AR process of order one (AR1). This process is a sequence (1) where: 11 { y t } ∞ t = 0 defjned by: y t = ρ y t − 1 + e t • Initial value y 0 is known • e t is an i.i.d. innovation that follows e t ∼ N ( 0 , σ 2 e ) Stability condition : | ρ | < 1

Vector Autoregressions (VARs) Group elements as follows (2) A VAR is a model with n variables with k lags. 12 Example: two variables { y 1 , y 2 } , one lag y 1 , t = φ 11 y 1 , t − 1 + φ 12 y 2 , t − 1 + e 1 , t y 2 , t = φ 21 y 1 , t − 1 + φ 22 y 2 , t − 1 + e 2 , t � � � � � � φ 11 φ 12 Y t = Φ = e t = y 1 , t e 1 , t φ 21 φ 22 y 2 , t e 2 , t and rewrite the model in matrix form as Y t = Φ Y t − 1 + e t

Vector Autoregressions (VARs) A VAR is a model with n variables with k lags. (2) and rewrite the model in matrix form as 12 Group elements as follows Example: two variables { y 1 , y 2 } , one lag y 1 , t = φ 11 y 1 , t − 1 + φ 12 y 2 , t − 1 + e 1 , t y 2 , t = φ 21 y 1 , t − 1 + φ 22 y 2 , t − 1 + e 2 , t � � � � � � φ 11 φ 12 Y t = Φ = e t = y 1 , t e 1 , t φ 21 φ 22 y 2 , t e 2 , t Y t = Φ Y t − 1 + e t

Vector Autoregressions (VARs) Group elements as follows (2) A VAR is a model with n variables with k lags. 12 Example: two variables { y 1 , y 2 } , one lag y 1 , t = φ 11 y 1 , t − 1 + φ 12 y 2 , t − 1 + e 1 , t y 2 , t = φ 21 y 1 , t − 1 + φ 22 y 2 , t − 1 + e 2 , t � � � � � � φ 11 φ 12 Y t = Φ = e t = y 1 , t e 1 , t φ 21 φ 22 y 2 , t e 2 , t and rewrite the model in matrix form as Y t = Φ Y t − 1 + e t

Vector Autoregressions (VARs) interpretation. • k -periods ahead: • Two -periods ahead: VAR models of this form Reduced-form models are useful for forecasting: • One -period ahead: are often referred to as reduced-form models: (3) 13 Y t = Φ Y t − 1 + e t • Parameters Φ do not have any intrinsic economic meaning. • Innovations e t do not have an economic or structural E t [ Y t + 1 ] = Φ E t [ Y t ] + E t [ e t + 1 ] = Φ Y t E t [ Y t + 2 ] = Φ E t [ Y t + 1 ] = ΦΦ Y t E t [ Y t + k ] = Φ · · · Φ Y t = Φ k Y t

Vector Autoregressions (VARs) interpretation. • k -periods ahead: • Two -periods ahead: VAR models of this form Reduced-form models are useful for forecasting: • One -period ahead: are often referred to as reduced-form models: (3) 13 Y t = Φ Y t − 1 + e t • Parameters Φ do not have any intrinsic economic meaning. • Innovations e t do not have an economic or structural E t [ Y t + 1 ] = Φ E t [ Y t ] + E t [ e t + 1 ] = Φ Y t E t [ Y t + 2 ] = Φ E t [ Y t + 1 ] = ΦΦ Y t E t [ Y t + k ] = Φ · · · Φ Y t = Φ k Y t

Identifjcation and causality

... but what caused the innovation e 1 t to jump in the fjrst place? Innovations and dynamic responses VAR models are useful to predict the dynamic response of a set of Thus, we compute Impulse Responses Functions (IRFs): This gives us a clear picture of the dynamic response of variables ... 14 macroeconomic variables to a sudden change in one of the variables. 1. The innovation e 1 , t increases by one percent in period, t = 1. 2. The variable y 1 , t immediately responds in the fjrst period, t = 1. 3. Variables y 1 , t and y 2 , t respond in the subsequent periods, t > 1.

Innovations and dynamic responses VAR models are useful to predict the dynamic response of a set of Thus, we compute Impulse Responses Functions (IRFs): This gives us a clear picture of the dynamic response of variables ... 14 macroeconomic variables to a sudden change in one of the variables. 1. The innovation e 1 , t increases by one percent in period, t = 1. 2. The variable y 1 , t immediately responds in the fjrst period, t = 1. 3. Variables y 1 , t and y 2 , t respond in the subsequent periods, t > 1. ... but what caused the innovation e 1 , t to jump in the fjrst place?

Structural shocks and contemporaneous responses If we want some economic insight on the response of variables, we meaning: variable To resolve this issue we need to impose some structure on the model. 15 need to identify structural shocks ε i , t . • A structural shock ε i , t is likely to impact more than one variable y i , t contemporaneously ! • The effect of the shock will show up in the residuals e i , t but we cannot say if this is coming from ε 1 , t or ε 2 , t This is why movements in the residuals e i , t are without economic • A movement in e 1 , t is not necessarily due to a shock to the y 1 , t • It might be the contemporaneous by-product of a shock to y 2 , t or to both variables y 1 , t and y 2 , t

Structural shocks and contemporaneous responses If we want some economic insight on the response of variables, we meaning: variable To resolve this issue we need to impose some structure on the model. 15 need to identify structural shocks ε i , t . • A structural shock ε i , t is likely to impact more than one variable y i , t contemporaneously ! • The effect of the shock will show up in the residuals e i , t but we cannot say if this is coming from ε 1 , t or ε 2 , t This is why movements in the residuals e i , t are without economic • A movement in e 1 , t is not necessarily due to a shock to the y 1 , t • It might be the contemporaneous by-product of a shock to y 2 , t or to both variables y 1 , t and y 2 , t