Download The Contemporaneous Correlation Between

The Contemporaneous Correlation Between
Price and Output Shocks
George K. Davis
Dept. of Economics
Miami University
Oxford, Ohio 45056
Bryce E. Kanago
Dept. of Economics
Miami University
Oxford, Ohio 45056
June, 1999
Abstract
In the past decade controversy arose concerning the cyclical behavior of prices. An issue that had
appeared settled for many years was re-opened when Kydland and Prescott (1990) and Cooley
and Ohanian (1991) presented evidence that prices were counter-cyclical. This evidence has
stood up, but its interpretation has varied. Hall and Judd and Tehran argued that the
contemporaneous correlation between output and prices reflects the source of the current shock
as well as the nature of the adjustment process from short-run to long-run equilibrium; and the
adjustment to the long-run imparts a bias toward finding a negative price-output correlation. In
this paper we control for the long-run adjustment in a general way and find evidence that supports
the HJT hypothesis. Indeed, for the U.S. controlling for the long-run adjustment switches the
correlation of the shocks from negative to positive. Controlling for the long-run adjustment alone
is insufficient to reveal the source of shocks. However, if in the initial period prices are equally
sticky in response to demand and supply shocks, then the correlations do have something to say
about the nature of the disturbances. Under the assumption of equal initial price stickiness, the
data suggest that economies are buffeted by both demand and supply shocks in about equal
portions.
Sharp attention has recently been re-focused on the correlation between the price level and
aggregate output. For many years it was well accepted that prices were procyclical. Lucas
(1977), following Burns and Mitchell (1946), took procyclical prices as one of the facts that
models of the business cycle should explain. The popularity of the short-run Phillips curve
reflected this widespread acceptance of procyclical prices. Kydland and Prescott (1990) and
Cooley and Ohanian (1991) challenged this interpretation of the data. These authors argued that
for the post-Korean War era the price level has been countercyclical. Wolf (1990) suggested that
this negative correlation was primarily the result of the oil price shocks in the 1970s. Ravn and
Sola (1995) studied this question with a version of Hamilton 's switching regression methodology
to investigate shifts in the correlation between prices and output. They concluded that in general
the negative correlation is robust to changes in regimes, including regimes that are not
characterized by oil price shocks.
Empirical evidence on the correlation between prices and output was used to differentiate
between competing explanations of economic fluctuations. In their 1986 text Hall and Taylor
wrote “(a) fact of the business cycle that is missed in the classical models is the positive
correlation between prices and output.” Parkin and Bade (1992) wrote “There is also a generally
positive correlation between inflation and economic activity. In the mainstream economic models
we’ve studied, these correlations arise from the fact that money is one of the variables influencing
the economy.” On the other hand, Kydland and Prescott (1990) in their study of the cyclical
behavior of money and prices concluded “that any theory in which procyclical prices figure
crucially in accounting for post-war business cycle fluctuations is doomed to failure.”
Hall (1995) and Judd and Trehan (1995), henceforth HJT, questioned the value of the
price-output correlation as evidence on competing explanations of the business cycle. They
argued that long-run adjustment of prices and output masks the nature of shocks to the economy.
In a textbook AD-AS model with a flat short-run supply curve, an increase in aggregate demand
increases output but not the price level. However, as prices adjust upward, output will fall back
toward its long-run equilibrium creating a negative price-output correlation. Similarly, the
reversion to equilibrium from a temporary supply shock also imparts a bias toward finding a
negative price-output correlation. This dynamic adjustment could create a negative correlation
2
between prices and output even if most shocks were from the demand side. We refer to this bias
introduced by dynamic adjustment as the HJT hypothesis.
In this paper we identify the contemporaneous correlations between price and output
shocks by examining the residuals from reduced form price and output equations. Correlations
between macroeconomic variables and correlations between their shocks both provide useful
descriptions of the macroeconomy. However, we focus on correlations between shocks to
evaluate the HJT hypothesis. For the U.S. we find that the correlation of the contemporaneous
price and output shocks is positive even though the correlation using the Hodrick-Prescott filter
or first differencing is negative. In contrast to the U.S., the correlation between residuals from
reduced forms is negative in Canada and the U.K. Nevertheless, the correlation is typically
greater for the reduced form residuals than it is for the correlation obtained with the detrended
data. Thus, we find support for the HJT hypothesis.
The correlations between the reduced form residuals do not allow us to identify the shocks
as coming primarily from either the demand side or the supply side. However, if the short-run
response of prices to demand shocks relative to supply shocks is known, then some progress can
be made in assessing the importance of demand shocks relative to supply shocks. Menu cost
theory and a simple model due to Rotemberg (1996) give reason to assume that the impact effect
on prices from demand and supply shocks will be the same. Conditional on this assumption, we
find that the U.S., U.K., and Canada experience roughly the same number of shocks from both
sides.
In the next section we review HJT’s argument that the price-output correlations reflect the
effects of both shocks and the long-run adjustment to earlier shocks. In section 3 we describe the
data used to estimate the model and we present our results on the HJT hypothesis in section 4. In
section 5 we discuss the interpretation of the correlation. We conclude in section 6.
2.
The HJT Effect
In this section we show the general nature of the issues raised by the HJT analysis and in
the process introduce some necessary notation. We describe the economy by specifying the
endogenous variables, the exogenous variables, the contemporaneous demand and supply shocks,
e dt and e st respectively, and the relationships between the variables and the shocks. It is an
3
abstraction to think of only two shocks, but it is traditional to proceed in this way and we may
interpret e dt and e st as aggregate shocks comprised of many disaggregate disturbances. We
assume that the shocks are independent white noise, but otherwise specify things in a general way.
Let Yt be the level of real output, Pt be the price level, Zt be a vector of endogenous
variables, and Xt be a vector of exogenous variables. The economy is defined by the system of
implicit equations:
F1 (Yt , Pt , Zt , Xt , Yt-1 , Pt-1 , Zt-1 , Xt-1 , ..., e dt , e st) = 0
F2 (Yt , Pt , Zt , Xt , Yt-1 , Pt-1 , Zt-1 , Xt-1 , ..., e dt , e st) = 0
F3 (Yt , Pt , Zt , Xt , Yt-1 , Pt-1 , Zt-1 , Xt-1 , ..., e dt , e st) = 0
where F3 is a vector of functions. The lagged variables are included to capture any endogenous
dynamics in the model and the information value of these variables in the formation of
expectations.
We now think of solving this system for the equilibrium values of the endogenous
variables and then linearizing the results for price and output. This solution yields equations of
the form
Y t = A(L)Y t + B(L)P t + C(L)Z t + D(L)X t + u t
y
P t = Y(L)Y t + W(L)P t + L(L)Z t + G(L)X t + u t
p
(1)
(2)
p
where A(L) and the like are polynomial lag operators, and the price and output shocks, u t
y
and u t , are linear combinations of the demand and supply shocks.
The structure of this solution reveals the problem identified by HJT. To illustrate the
nature of this problem, ignore detrending issues, assume each variable has zero mean, and
suppose that only one lagged value of Yt , Pt , and Zt enter equations (1) and (2). In this case the
covariance between Yt and Pt is given by
4
EY t P t = E(aY t−1 + bP t−1 + cZ t−1 + u t )(yY t−1 + zP t−1 + kZ t−1 + u t )
y
EY t P t = h 1 r 2Y + h 2 r 2P + h 3 r 2Z + h 4 r ZP + h 5 r YZ + r u Yu P
p
(3)
where h 1 = k * (ay), h 2 = k * (bz), h 3 = k * (ck), h 4 = k * (bk + cz), h 5 = k * (ak + cy), and
k = 1/(1 − az − by). First, note that if there are no dynamics, the coefficients on the lagged
variables are all zero, so the θ's are all zero, and the covariance of Y and P is just the covariance
of the shocks uP and uY in the two equations. However, once dynamics are allowed, we see from
equation (3) that the covariance of Y and P does not depend only on the covariance of the shocks
uP and uY . The covariance of Y and P also depends on the variances of Y, P, and Z; the
covariances of Y and Z, and P and Z; and the values of the coefficients. Hall (1995) and Judd and
Tehran (1995) provide examples of specifications and sets of parameters where the covariance of
the contemporaneous price and output shock, r u yu p , is positive, while the covariance between
observed price and output, EYt Pt, is be negative.
Detrending methods that only render Y and P stationary do not remove effects of long-run
adjustments. For example, first-differencing would mean the variances and covariances in
equation (3) are variances and covariances of first differences rather than levels. It does not make
the covariances zero. To separate the shocks from the effects of long run adjustments up and uy
must be estimated directly.
To estimate up and uy we must first decide whether to estimate the model in levels or in
first-differences. Chada and Prasad (1994) raised a closely related question. They asked whether
it was appropriate to examine the correlation between output and the price level or the correlation
between output growth and inflation. This question is nontrivial since with their data the sign of
the correlation depends on the answer. There does not appear to be a clear theoretical reason to
prefer one correlation over the other. We estimate equations (1) and (2) in levels since the levels
specification does not impose any restrictions on the estimated coefficients and is consistent with
a specification in growth rates. The OLS estimates of the coefficients will be consistent and so
yield consistent estimates of the residuals.1
3.
1
Data
See Hamilton (1994) p.553. Doan (1992) p. 8-3 recommends estimating VARs in level as a
general matter.
5
The variables included for the United States specification are the natural logs of real GDP,
the GDP deflator, the relative price of oil, M1, and the S&P 500; and the levels of the federal
funds rate, the ratio of net exports to GDP, the spread between the three month treasury bill rate
and ten year treasury bond rate, and the ratio of defense spending to GDP. For the U.S.,
consistent measures of the money supply begin in 1959 and so does our sample. The import and
export data and the rates on treasury securities come from International Financial Statistics (on
CD-rom). The remaining data comes from Citibase.
The relative price of oil is included because several economists have focused on its role in
recessions (see Hamilton (1983) for example). The federal funds rate and M1 are indicators of
monetary and credit policy. The S&P 500 is included to proxy for changes in the productivity of
capital (see, for example, Barro and Sala-i-Martin (1990)). The ratio of defense spending to GDP
is a measure of fiscal policy. The spread between short and long-term rates of interest proxies for
conditions in credit markets and the ratio of net exports to GDP proxies for disturbances to the
international sector. In general, we thought it was better to begin with a generous specification
since omitting a relevant variable would fail to account for long-run adjustment. We find below
that the results are similar for a more parsimonious specification.
For Canada and the United Kingdom our data comes exclusively from International
Financial Statistics and our sample begins in 1957. The variables included for Canada and the
U.K. are the natural logs of real GDP, the GDP deflator, narrow money, and the price of oil; the
levels of the interest rate on three month government bills, the spread between long and
short-term government securities, the ratio of government consumption to GDP, and the ratio of
net exports to GDP. To generate the price of oil we converted the U.S. price of oil into Canadian
or British currency. All of the data is quarterly and the general specification for each country
includes eight lags of each variables in both the price and output equations.2
4.
2
The Empirical Results on the HJT Hypothesis
We checked the robustness of the estimates for each equation. In general CUSUM tests showed no
instability and Breusch-Pagan LM tests showed no serial correlation in the residuals. The exceptions were
that the parsimonious specification for the U.S. price equation and the full specification for the U.K. price
equation exhibited some serial correlation. Our results did not significantly change when we corrected for
serial correlation so we report the original results. There was some evidence of instability in the U.S. price
equation in the early 1980s, but it was small and temporary.
6
Earlier work detrended price and output series by either first-differencing or by using the
Hodrick-Prescott filter. The HJT hypothesis holds that price-output correlations with these filters
will be smaller than the correlation between the contemporaneous price and output shocks. The
correlation between the detrended series for both of these filters are given in lines one and two in
Table 1.3
Table 1
Contemporaneous Price-Output Correlations
filter
Hodrick-Prescott
first-differences
full specification
sparse specification
U.S.
-0.53
-0.15
0.12
0.03
Canada
-0.49
-0.14
-0.3
-0.32
U.K.
-0.6
-0.27
-0.16
-0.22
notes: the sparse specification includes eight lags of output and the price level only.
The correlations using the Hodrick-Prescott filter or first-differencing yield the typical
negative values for all three countries and so our data reproduces the results from earlier studies.
Line three in Table 1 shows the correlation of the price and output residuals that were estimated
using the full specification for each country. For the U.S. the correlation between the
contemporaneous price and output shocks is positive providing strong support for the HJT
hypothesis. The U.S. data is thus consistent with a positive correlation between contemporaneous
shocks, but a negative correlation between detrended price and output. The correlation between
contemporaneous shocks in the full specification is negative for both Canada and the United
Kingdom, but is larger than the correlation with detrended price and output with the exception of
Canada when first-differencing is used to detrend price and output. So, in five of six possible
cases the HJT hypothesis holds.
3
Our approach is similar to the methodology of den Haan (1996). He estimates a VAR to control
for dynamic effects, and examines the correlations between the forecast errors for prices and output at
different horizons. Our results complement den Haan’s in that we also find a small positive correlation
between contemporaneous price and output shocks in the U.S. They differ from den Haan’s in that we
examine data from Canada and the U.K. and find a negative correlation between contemporaneous price
and output shocks. We also study the period-by-period errors in all three countries and find for each
country that both supply and demand shocks contribute significantly to short-run fluctuations; while den
Haan concludes that demand shocks are the most important shocks for short-run fluctuations.
7
To check the robustness of our results, we present in line four the results from
specifications that only included lagged values of price and output. This sparse specification
yields qualitative results that are identical to the general specification.
5.
Interpretation of the Correlation of the Residuals
a.
difficulties in identifying the nature of a shock
Estimating the contemporaneous price and output shock allows us to directly address the
HJT hypothesis, but does not by itself allow us to identify the underlying demand and supply
shocks. To see why recall that the price and output shocks are linear combinations of the demand
and supply shocks, and can be written
u t = ae dt + be st
y
u t = ae dt + be st ,
p
where the coefficients on the right-hand-side disturbances measure the impact effects on output or
price of a unit shock. Positive demand shocks raise both price and output in the current period,
so we take a and α to be positive. On the other hand, positive supply shocks raise output, but
cause prices to fall; so we take b to be positive, but β to be negative. The covariance of price and
output implied by our model is given by
r u yu p = E[u Y * u P ] = aar 2d + bbr 2s
Given the signs of the impact effects, the coefficient on the variance of the demand shock should
be positive, while the coefficient on the variance of supply shocks should be negative.
Since the covariance depends on units, its absolute size is not important. So, rewrite the
covariance as
r u yu p = aar 2d (1 + (b/a)(b/a)(r 2s /r 2d ))
8
The sign of the covariance depends on three relative magnitudes. The ratio of the variances is a
measure of the relative contributions of the two types of shocks to price and output fluctuations.
To take an extreme example suppose there are no supply shocks at all. In this case the variance
of supply shocks is zero and σYP will be positive. More generally, if demand shocks account for a
larger share of fluctuations than supply shocks, then r u yu p is more likely to be positive. The ratio
(b/a) is a measure of the relative impact effects of demand and supply shocks on output. If
demand shocks have large effects on output relative to supply shocks; then the ratio (b/a) will be
small, and r u yu p will tend to be positive. The combined ratio(b/a)(r 2s /r 2d ) is a measure of the
relative contributions to output fluctuations of supply and demand shocks. If this term is large,
then supply shocks are relatively important, and the covariance r u yu p will tend to be negative; but
if it is small, then demand shocks are relatively stronger andr u yu p will tend to positive.
The remaining ratio is (β/α). We may interpret this ratio as a measure of the initial
stickiness of prices in response to demand shocks relative to initial price stickiness in response to
supply shock. So, for example, if prices are very sticky in response to demand shocks and quite
flexible in response to supply shocks, this ratio is large and it is more likely that the covariance is
negative. Indeed, the covariance between contemporaneous price and output shocks can be
negative even if the variance and output effect of demand shocks were larger than those of supply
shocks. Hence, without further restrictions the covariance alone does not tell us very much about
the relative importance of demand and supply shocks.
We are not aware of any evidence on this issue, so we turn to theory for guidance.4 The
modern theory of price stickiness relies on menu costs arguments and these arguments rely, in
turn, on the envelope theorem.5 As a result, the menu cost argument weighs equally on shocks
from the demand and supply sides, so the initial price response is the same for both shocks; indeed
in the base case there is no price response at all to either a demand or supply shock. Rotemberg
(1996) constructs a model with costly price adjustment that is subject to both demand and supply
shocks. In his model the impact effect on prices from a demand shock is larger than the impact
4
There has been work done on the asymmetry of the price response to positive and negative demand
shocks (for example, see Cabellero and Engle (1992) or Ball and Mankiw (1994)). However, here we are
concerned with a comparison of price response to demand and supply shocks.
5
See, for example, Akerloff and Yellen (1985).
9
effect on prices of a supply shock. This would mean that (β/α) is less than one and would impart
a bias toward finding a positive correlation between contemporaneous price and output shocks.
In Rotemberg’s model this difference in price response does not come from the price-setting
structure, but instead comes from the assumption that the autoregressive nature of money and
technology growth differ. For example, if money and technology were both modeled as random
walks, the price response to both shocks would be the same. We now provisionally take the
magnitude of the ratio (β/α) to be unity. Of course, this can occur if prices are very sticky since
both α and β can be small.6
With the equal stickiness assumption, the sign of the covariance between prices and output
is determined by the condition
r u yu p m 0
if and only if
ar 2d m br 2s .
A positive covariance is now evidence of a relatively strong output effect from aggregate demand,
a large variance of demand shocks, or both. On the other hand, if the output effect from supply
shocks, the variance of supply shocks, or both are relatively large; then the covariance will be
negative.
b.
sample correlation
We have already discussed the sample correlations in evaluating the HJT hypothesis, and
we return to them now in the present context. The sample correlation coefficient between the
forecast errors in the price and output equations for the U.S. is positive, but relatively small at
.12. This value suggests that demand shocks are relatively more important in the U.S. than are
supply shocks. The sample correlations for the U.K. and for Canada are both negative and larger
in absolute value than the correlation for the U.S. These results suggest that in the U.K. and
Canada supply shocks are relatively more important.
The sign of the correlation depends on both the frequency and size of shocks. A positive
correlation may occur for two reasons. First, it could be the case that most of the shocks to the
economy are demand shocks. An alternative is that the economy is buffeted by about equal
quantities of demand and supply shocks, but a few of the demand shocks are large. Similarly,
6
The adoption of an explicit identifying assumption also differentiates us from den Haan (1996).
10
negative correlations may arise because economies are pestered by supply shocks, or because
there are a mix of supply and demand shocks with a few large shocks coming from the supply
side.
c.
quarter-by-quarter cross products
To consider these two possibilities we plot quarter-by-quarter cross-products of the
residual estimates from the two equations for each of the countries. The shaded areas in each
graph are recessions. If a prevalence of demand shocks is responsible for the correlation in the
U.S., then most of the cross-products should be positive. On the other hand, if both types of
shocks disturb the economy, then we should observe at least some large positive and some large
negative cross-products. An analogous argument holds for the U.K. and Canada. If in the U.K.
and Canada the negative correlation is generated by a long series of supply shocks, then most of
the cross-products should be negative; but if these two countries are buffeted by both demand and
supply shocks, then we should see some large positive and some large negative cross-products.
It is clear from the figures that each country is subject to both demand and supply shocks.
Let us take up the case of the U.S. in some detail. If we define large by greater than one standard
deviation, then there are twelve large demand and ten large supply shocks.7 The mid-1960s
appear to be dominated by variations in demand. The correlation coefficient in the sub-sample
from 1964:1 to 1968:4 is .21. On the other hand, the 1970s appear to be dominated by supply
shocks. The OPEC recession of 1974 is associated with large supply shocks and the correlation
coefficient in the sub-sample from 1970:1 to 1977:4 is -.25. The recession in 1970 also appears
to be associated with a large supply shock, but large positive cross-products accompany the 1980
and 1960 recessions.8 Small cross-products characterize the 1982 and 1990 recessions.
7
The largest cross-products are in: 1960, 1970, and 1978. Two of these are positive and one is
negative. The first two are associated with recessions. The third is the result of the Blizzard of 1978 and a
concurrent sharp increase in the price of food. The 1979 Economic Report of the President assigns the
extraordinary growth in the second quarter of 1978, which was over 12% and is the largest for any quarter
in the sample, to the recovery from the blizzard. In the same quarter inflation was over 10%, the second
highest rate in our sample. Much of this price rise was fueled by a 20% increase in the price of food.
8
It is interesting to note that 1970 contained an energy price shock, a ten week strike by General
Motors, and the disruptive influences of downsizing in defense and the bankruptcy of the Penn Central.
See Hamilton (1983) and the Economic Report of the President (1971) p. 23.
11
Figure 1: Cross-Products for the United States
0.00010
0.00005
0.00000
-0.00005
straight horizontal lines are
one standard deviation bands
-0.00010
60
65
70
75
80
85
90
95
The story is similar for the U.K. and Canada. There are fewer large shocks for Canada
than for the U.S. and U.K.; but these large shocks remain roughly equally divided between supply
and demand shocks. The cross-product of the residuals for the U.K. are plotted in Figure 2. We
see both large positive and negative cross-products suggesting that the U.K. economy, like the
U.S., is subject to significant demand and supply shocks. The 1970s are again a time of apparent
supply shocks with the correlation coefficient over this decade equal to -.23. The protracted
recession in the early 1980s is accompanied by small cross-products.9 The pattern in the 1960s
and 1970s is again similar for Canada.
In sum, the examination of the cross-products of the residuals indicates that there have
been substantial shocks of both types in about equal frequency. The cross-products also tell a
story broadly consistent with a general reading of recent macroeconomic history. We again
checked for the sensitivity of our specification by estimating the price and output equations with
only lagged values of prices and output. The correlation of the cross-products across the
specifications is .61 and again the pattern is similar in both cases.
9
For the U.K. and Canada we define the peak of an expansion as the quarter that began at least two
consecutive quarters of negative growth. We define a trough as the quarter in which two consecutive
quarters of positive growth began.
12
Figure 2: Cross-Products for the United Kingdom
0.0002
0.0001
0.0000
-0.0001
straight horizontal lines are
one standard deviation bands
-0.0002
-0.0003
60
6.
65
70
75
80
85
90
95
Conclusion
The correlation between price and output has been used to judge the source of shocks.
Hall, and Judd and Trehan questioned this interpretation because the dynamic adjustment of prices
Figure 3: Cross-Products for Canada
0.00010
0.00005
0.00000
-0.00005
-0.00010
-0.00015
60
65
70
75
80
85
90
95
and output mask the origin of a shock. In particular, their arguments imply that the long-run
adjustment imparts a bias towards finding a negative price-output correlation. We used reduced
form equations to estimate price and output shocks; and found support for their hypothesis. With
13
our approach the correlations are typically small, ranging from .12 in the U.S. to -.33 for the
sparse specification for Canada.
Accounting for long-run adjustment does not, by itself, allow the sign of the price-output
correlation to identify the primary source of macroeconomic shocks. To identify the shocks we
impose the assumption that the impact effect on prices is the same magnitude for supply and
demand shocks. With this assumption we examine the cross-product of the residuals from the
price and output equations to get a view of the nature of the quarter-by-quarter shocks. An
inspection of these cross-products suggests that shocks come from both sides in about equal
proportions. Nevertheless, some periods happen to be dominated by supply shocks while others
are driven primarily by demand shocks.
Cochrane (1994) found that none of the usual suspects: money, technology, oil shocks, or
credit shocks can robustly account for the bulk of economic fluctuations. Temin (1998), using a
very different approach, also arrived at the eclectic conclusion of varied sources of shocks. Our
results support this conclusion that different types of shocks, some easily observable and some
not, some dominating of some periods and some others, conspire to produce the business cycle.
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16
Summary Table
United States
Lagged prices and output
General Specification
y-equation
p-equation
y-equation
p-equation
lm(1)
0.21
0.17
0.16
0
dickey-fuller
-5.69
-4.96
-5.05
-4.9
yes
some instability
beginning mid-1980s
a bit in early1980s
yes
stable
correlation
0.12
0.08
correlation
across specs
0.59
United Kingdom
General Specification
Lagged prices and output
y-equation
p-equation
y-equation
p-equation
lm(1)
0.18
0.05
0.98
0.53
dickey-fuller
-4.84
-4.54
-5
-4.52
yes
yes
yes
yes
stable
correlation
-0.22
-0.15
correlation
across specs
0.61
Canada
General Specification
Lagged prices and output
y-equation
p-equation
y-equation
p-equation
lm(1)
0.69
0.89
0.86
0.43
dickey-fuller
-5.09
-4
-4.65
-3.46
yes
yes
yes
yes
stable
correlation
correlation
across specs
-0.3
-0.33
.54
notes: The entry in the lm column are p-values. The dickey-fuller stats are 't" stats
Stable means that the plot of CUSUM and CUSUMSQ stay within the 5%
bounds.