5.2 Historical decomposition of the business cycle
Using historical decompositions, we also estimate the individual contributions of each structural shock to the movements in GDP, inflation and the long-term interest rate over the sample period. The historical decompositions of each variable into the estimated structural shocks are calculated as follows:
- The VAR(3) model is written in companion form[5] (ie, as a VAR(1) model) as,
- Using backward substitution and the Wald decomposition (see Enders (2004) for details), the model variables at each point in time (Yt) can be represented as a function of initial values (Y0) plus all the structural shocks of the model
The historical decomposition of the shocks to GDP is shown in Figure 13, which is discussed below.
In the late-1980s, government spending subtracts from output while net taxes contribute positively. Interest rate shocks play a modestly dampening role on GDP. GDP's own shocks on itself and inflation shocks contribute positively to GDP.
In the recession of the early 1990s, neither government spending nor net taxes shocks make any material contribution to output. Interest rate shocks and inflation shocks dampen output, but GDP's own shocks are by far the most significant component of the negative shocks to output.
During the recovery period of the mid-1990s, government spending makes a very small positive contribution to output, whereas net taxes play a dampening role. During the 1998 growth slowdown, which coincided with the Asian economic crisis, both government spending and net taxes make a negative contribution to output.
Over 2000 to 2007, GDP rises to levels well above trend. Net taxes make a strongly positive contribution to the output gap over 2000 to 2005, before becoming negative in their contribution. Government spending makes a minimal contribution over 2000 to 2003, but then makes a positive contribution over 2004 to 2007. Shocks from the long-term interest rate dampen GDP over 2000 and 2001, and thereafter make a positive contribution. Inflation shocks appear to make a minimal contribution to the output cycle over this period. GDP's own shocks on itself explain a significant portion of the positive output gap over 2003 to 2007.
In 2008, the economy entered recession and the output gap turned negative in 2009. Government spending makes virtually no contribution to the downturn over this period, whereas net taxes make a negative contribution. Interest rate and inflation shocks make a positive contribution, partially offsetting the large negative contribution from GDP's own shocks. The negative contribution from net taxes is counter-intuitive since there were substantial permanent tax cuts delivered in late 2008. This may be partly due to the counter-intuitive result in section 4 that net tax shocks have a negative output multiplier at certain horizons or the lagged effects of previous positive shocks to net taxes.
- Figure 13: Historical decomposition of GDP
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- Source: Authors' calculations
The historical decomposition of the shocks to inflation in the GDP deflator rate is shown in Figure 14.
In the 1980s, a period in which inflation is mostly above trend, net taxes and government spending contribute somewhat positively to inflation shocks. GDP shocks also add to inflation during the late 1980s. Interest rates and inflation shocks on itself (ie, residual) have contributions which change sign throughout the decade.
In the 1990s, net taxes generally dampen inflation while government spending plays a very minor role in adding to inflation. In the first half of the decade, GDP and interest rate shocks also subtract from inflation. Later in the 1990s, net taxes dampen inflation along with interest rates and inflation shocks on itself.
In the 2000s, net taxes add to inflation in the first half of the decade and then make a negative contribution in the latter half. Government spending shocks partially offset this by making a negative contribution in the first half of the decade and a positive contribution in the latter half. GDP shocks make a positive contribution throughout the decade. Interest rate shocks play a dampening role in the first half of the decade, and add modestly to inflation in the latter half.
- Figure 14: Historical decomposition of inflation (GDP deflator)
- Source: Authors' calculations
- Figure 15: Historical decomposition of interest rate (10-year bond yield)
- Source: Authors' calculations
- [5]Any stationary VAR(P), where P>1, can be rewritten as a VAR(1) by constructing the companion form. This allows the statistical properties of any VAR(P) to be directly computed using only the results of a VAR(1).
The historical decomposition of the shocks to the long-term interest rate (10-year government bond yield) is shown in Figure 15.
Over the mid to late 1980s, net taxes make a positive contribution to interest rate shocks. Government spending shocks are positive in the mid-1980s but become a modest dampening factor in the late 1980s. GDP's own shocks and own interest rate shocks also make a significant positive contribution to interest rate dynamics.
In the 1990s, total shocks to the interest rate are generally negative. Government spending shocks have a small positive effect on the interest rate in the middle of the decade. Net tax shocks initially make minimal contribution, but make a material negative contribution in the latter half of the decade. GDP and inflation dampen interest rates in the early half of the decade. Interest rate shocks on itself contribute negatively in the early 1990s, add positively to the middle of the decade, and then dampen again in the late 1990s.
In the first half of the 2000s, net tax shocks and GDP shocks contribute positively to the interest rate shocks. This is offset by negative contributions from government spending and interest rate shocks on itself. In the second half of the 2000s, government spending shocks contribute positively to the interest rate while net taxes dampen the interest rate. GDP shocks and interest rate shocks also contribute negatively over this period.
