Econ226_IVAC

# Econ226_IVAC - IV Markov-switching models A Introduction to...

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1 IV. Markov-switching models A. Introduction to Markov-switching models Measured as 100 times natural log (means vertical change of 10 is approximately a 10% increase) US real GDP, 1947-2008 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 725 750 775 800 825 850 875 900 925 950 cumulative drop, 2007:Q3 to 2009:Q1 = 2.4% US real GDP, 1947-2008 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 725 750 775 800 825 850 875 900 925 950

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2 Density of expansions 0 0.05 0.1 0.15 - 1 5 - 1 3 - 1 1 - 9- 7- 5- 3- 1135791 1 1 3 1 5 GDP growth 6 = 4.7 @ = 3.5 Density of expansions 0 0.05 0.1 0.15 -15 -12 -9 -6 -3 0 3 6 9 12 15 GDP growth Density of recessions 0 0.05 0.1 0.15 -15 -12 -9 -6 -3 0 3 6 9 12 15 GDP growth 6 = 4.7 @ = 3.5 6 = -1.2 @ = 3.5 Density of mixture 0 0.02 0.04 0.06 0.08 0.1 0.12 -15 -12 -9 -6 -3 0 3 6 9 12 15 expansion recession mixture
3 Density of mixture 0 0.02 0.04 0.06 0.08 0.1 0.12 -15 -12 -9 -6 -3 0 3 6 9 12 15 expansion recession mixture ) 2 | ( ) 2 Pr( ) , 2 Pr( ) , 2 Pr( ) , 1 Pr( ) , 2 Pr( ) ( ) , 2 Pr( ) | 2 Pr( = = = = = + = = = = = = t t t t t t t t t t t t t t t t S y f S y S y S y S y S y f y S y S Density of mixture 0 0.02 0.04 0.06 0.08 0.1 0.12 -15 -12 -9 -6 -3 0 3 6 9 12 15 Probability of recession 0 0.2 0.4 0.6 0.8 1 -15 -12 -9 -6 -3 0 3 6 9 12 15 GDP growth in quarter t Prob(NBER expansion in t + 1| NBER expansion in t ) = 164/174 = 0.94 Prob(NBER contraction in t +1 | NBER contraction in t ) = 35/44 = 0.80

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4 ± ± ± ±² ± ± ± ±³ ´ ± ± ± ± ´ ± ± ´ recessions of density from known 1 1 1 s for 0.06 2 s for 0.80 1 slide previous from known 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ) , , 2 | ( ) , | 2 Pr( ) | Pr( ) | , , 2 Pr( ) | , , 2 Pr( ) | , 2 Pr( ) | , 2 Pr( ) | , 1 Pr( ) | , 2 Pr( ) | ( ) | , 2 Pr( ) , | 2 Pr( t t t t t t t t t t t t t s t t t t t t t t t t t t t t t t t t t t t t t t y s S S y f y s S S y s S y y s S S y y s S S y y S y y S y y S y y S y y f y y S y y S = = = = = = = = = = = = = + = = = = = = + + = = + + + = + + + + + + + + + + + + + + + Probability of recession in t given y(t) = y 0 0.2 0.4 0.6 0.8 1 -15 -12 -9 -6 -3 0 3 6 9 12 15 y = GDP growth in quarter t Probability of recession in t+1 given y(t+1)=y(t)=y 0 0.2 0.4 0.6 0.8 1 -15 -12 -9 -6 -3 0 3 6 9 12 15 y = GDP growth in t+1 and t Probability of recession in t given y(t) = y 0 0.2 0.4 0.6 0.8 1 -15 -12 -9 -6 -3 0 3 6 9 12 15 y = GDP growth in quarter t Probability of recession in t+1 given y(t+1)=y(t)=y 0 0.2 0.4 0.6 0.8 1 -15 -12 -9 -6 -3 0 3 6 9 12 15 y = GDP growth in t+1 and t Prob = 0.6 at y = -2.2 Prob = 0.6 At y = -0.8
5 Can also calculate “smoothed probability” ) , | 2 Pr( 1 + = t t t y y S Nonparametric summary of NBER labels: ) 1 | ( = t t S y f estimated nonparametrically 80 . 0 ) 2 | 2 Pr( 1 = = = + t t S S 94 . 0 ) 1 | 1 Pr( 1 = = = + t t S S Density of expansions 0 0.05 0.1 0.15 -15 -12 -9 -6 -3 0 3 6 9 12 15 GDP growth Density of recessions 0 0.05 0.1 0.15 -15 -12 -9 -6 -3 0 3 6 9 12 15 GDP growth 6 = 4.7 @ = 3.5 6 = -1.2 @ = 3.5

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6 Could also approach parametrically: ) , ( ~ ) 1 | ( 2 1 σ µ N S y f t t = ) , ( ~ ) 2 | ( 2 2 N S y f t t = 11 1 ) 1 | 1 Pr( p S S t t = = = + 22 1 ) 2 | 2 Pr( p S S t t = = = + ± ± ± ±² ± ± ± ±³ ´ ± ± ± ± ± ± ´ ± ± ´ ) , ( 1 1 1 s for p11 2 s for p22 1 step previous from known 1 1 2 2 ) , , 2 | ( ) , | 2 Pr( ) | Pr( ) | , , 2 Pr( N t t t t t t t t t t t t t y s S S y f y s S S y s S y y s S S = = = = = = = = + + = = + + + ) | , 2 , 2 Pr( ) | , 1 , 2 Pr( ) | , 2 , 1 Pr( ) | , 1 , 1 Pr( ) | ( 1 1 1 1 1 1 1 1 1 t t t t t t t t t t t t t t t t t t y y S S y y S S y y S S y y S S y y f + + + + + + + + + = = + = = + = = + = = = 0.74 0.80 p 22 0.92 0.94 p 11 3.35 3.47 @ -0.57 -1.19 6 2 4.62 4.73 6 1 Estimated from GDP alone Estimated from NBER dates and GDP Parameter
7 Full-sample smoothed probabilities and NBER recessions 1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 Current filter probabilities and NBER recessions 1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 Current filter probabilities and NBER recessions 1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002 -0.25 0.00 0.25 0.50

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Econ226_IVAC - IV Markov-switching models A Introduction to...

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