^{1}

^{*}

^{2}

This paper extends the method of regressing production and sales variables on a set of seasonal dummy variables and a linear trend for empirical tests on the production smoothing hypothesis of inventory investment. Unit root testing procedures form the basis for constructing the bootstrap confidence intervals for the ratio of the variance of production to the variance of sales. Models used to test for a unit root when there are structural breaks in the linear time trend are converted to equations comprising deterministic and stochastic components. A limiting distribution of the test statistic is derived. Bootstrap resampling is conducted over deterministic parts of the converted equations to construct confidence intervals for the relative variance ratio for G7 countries. Primary findings for these countries are production-counter smoothing phenomena.

This paper uses macroeconomic data from G7 countries related to production, sales, and inventory investment to test the production smoothing hypothesis of inventory investment of an earlier study [

Here and in the literature, G7 countries were chosen as the group of economically developed countries (Canada, France, Germany, the United Kingdom (UK), Italy, Japan, and the US), which were characterized by production counter- smoothing [

The paper is organized as follows. Section 2 presents descriptions of the analyzed data. Section 3 presents the unit root test models used for this study. Section 4 explains the econometric method used to construct the BCa bootstrap confidence intervals of the ratio of the variance of production to the variance of sales. Section 5 presents sketches of the proof that the presence of the deterministic seasonal dummy variables does not affect the limiting distribution for the test statistic in the equations to test for a unit root. The unit root test results for the production and sales of the G7 countries are presented. Section 6 presents and interprets the estimated BCa confidence intervals. Concluding remarks follow that section.

^{1}The database software Data stream is a global data platform that is used widely by universities and libraries such as Universiteit van Amsterdam, European University Institute, and Harvard Business School Baker Library. “It has historical, global coverage of equities, stock markets, commodities, and economic data. Information is updated daily.” (Baker Library Home page). Data stream provides post-1980 macroeconomic data for Japan; these data are constructed on the System of National Accounts 1993 (1993 SNA) basis. However, the Reuters Ecowin database provides Japanese data over a longer period, using the 1968 SNA basis. The Japanese data in this paper are kept on a 1968 SNAbasis.

Data used for this study were extracted primarily from the Datastream database (Thomson Inc.)1. Real quarterly time series for GDP and inventory investment, based on chain-linked price indices, were obtained for the US, the UK, and Canada. The US inventory investment, which is seasonally adjusted, is related only to the private sector. The Canadian GDP and inventory investment are also seasonally adjusted. The French and Italian price indices are expressed in terms of the fixed base method. France’s inventory investment is seasonally adjusted. The GDP deflator for Italy, which is not seasonally adjusted, is unavailable. Therefore, its producer price index, which is not seasonally adjusted, was used to deflate the Italian variables. The German nominal GDP and inventory investment, which are not seasonally adjusted, were deflated by the chain-linked GDP deflator, which is also not seasonally adjusted.

Seasonal variations are noteworthy when analyzing inventory fluctuations [

One report of the relevant literature [

The sample time periods differ slightly among related countries. The time periods analyzed in this study depend on whether significant results were obtainable for testing the variables for stationarity around the deterministic trend, with and without structural breaks.

^{2}The data deflator for Japan used here was constructed using the fixed base method.

^{3}Attempts to construct consistent data for West Germany on a quarterly basis were unsuccessful because the inventory investment figures represented on different volumes of the dataset were incompatible.

For Germany, the first differences of production and sales were tested for a unit root in augmented Dickey-Fuller (ADF) equations [

Canadian data were partitioned into two non-consecutive sample time periods. The 13-year time interval between the two observation periods was not included in our analyses because the unit root test outcomes showed no significant results for the sample periods that included this interval.

We use three models to test the production and sales time series for a unit root. Particular forms of the models presented in earlier reports [

(i) The ADF test with the linear time trend, that is

Δ y t = μ ^ + β ^ t + α ^ 1 y t − 1 + ∑ i = 1 k β ^ i Δ y t − i + e t (1)

where μ ^ , β ^ , α ^ 1 and β ^ ′ i are the parameters and where e t is the residual. The value of k is determined using the Akaike information criterion. Therein, Δ is the first difference operator.

(ii) The test of the model presented earlier [

y t = μ ^ + θ ^ D U t + β ^ t + γ ^ D T t + α ^ y t − 1 + ∑ i = 1 k β ^ i Δ y t − i + e t (2)

^{4}The truncation value for the t-statistic on β ^ i is 1.6 in absolute value. When the variables that are not seasonally adjusted are used in the ADF equations, this criterion is applied for the value of k. The maximum lag was eight, as reported by Zivot and Andrews [

where θ ^ and γ ^ are parameters, and where the value of k is determined as explained in earlier reports of the literature [

Let the break point be denoted as T B , then D U t = 1 for t ≥ T B + 1 , and D U t = 0 otherwise. D T t is defined such that D T t = t − T B for t ≥ T B + 1 , and D T t = 0 otherwise.

(iii) The test of another model presented earlier [

y t = μ ^ + θ ^ D T 1 t + β ^ t + γ ^ D T 2 t + α ^ y t − 1 + ∑ i = 1 k β ^ i Δ y t − i + e t (3)

D T j t is defined such that D T j t = t − T B j + 1 for t ≥ T B j + 1 , and D T j t = 0 otherwise, where j = 1, 2.

Following a procedure described in an earlier report [

(i) Converting Equation (1)

Letting α ^ = 1 + α ^ 1 , then Equation (1) can be written as

y t = μ ^ + β ^ t + α ^ y t − 1 + ∑ i = 1 k β ^ i Δ y t − i + e t , (1)'

which can be abbreviated further to

y t = μ ^ ′ + β ^ ′ t + O T ′ t , (1)"

where β ^ ′ = β ^ ( 1 1 − α ^ ) , μ ^ = μ ^ 1 − α ^ − β ^ α ^ ( 1 ( 1 − α ^ ) 2 ) , and

O T ′ t = 1 1 − α ^ L ( ∑ i = 1 k β ^ i Δ y t − i + e t ) 5 . Using Equation (1)" for the GDP and the sales

simultaneously, bootstrap resamplings on y t and t are made to run a series of ordinary least squares (OLS) regressions with the bootstrap samples on equation (1)". The variance of the O T ′ t term gives the variance of production for the GDP equation, and gives the variance of sales for the sales equation. Based on replications of the ratio of those variances, the BCa bootstrap confidence intervals can be constructed as explained in several reports of the relevant literature [

(ii) Converting Equation (2)

In rewriting Equation (2), terms ( 1 1 − α ^ L ) D U t and ( 1 1 − α ^ L ) D T t must be de- fined. Let D U C t = ( 1 1 − α ^ L ) D U t and D T C t = ( 1 1 − α ^ L ) D T t . Then it is straight- forward to demonstrate that D U C t = ( 1 − α ^ i 1 − α ^ ) for t = ( T B + i ) , i = 1 , 2 , ⋯ , N − T B , and D U C t = 0 for t < T B , and that D T C ( T B + i ) = ∑ j = 1 i [ D T i − ( j − 1 ) ] α ^ ( j − 1 ) for

i = 1 , 2 , ⋯ , N − T B , and D T C t = 0 for t ≤ T B . Here, N stands for the number of observations. Using these terms, Equation (2) can be rewritten as

y t = μ ^ ′ + θ ^ D U C t + β ^ ′ t + γ ^ D T C t + O T ″ t (2)'

In (2)', μ ^ ′ = μ ^ 1 − α ^ − β ^ α ^ ( 1 ( 1 − α ^ ) 2 ) , β ^ ′ = β ^ ( 1 1 − α ^ ) , and

O T ″ t = ( 1 1 − α ^ L ) ( ∑ i = 1 k β ^ i Δ y t − i + e t ) . Bootstrap resamplings are made with y_{t},

DUC_{t}, t, and DTC_{t} to apply a series of OLS regressions with these variables on Equation (2)' for the GDP and sales series to calculate the variances of O T ″ t . These give the variances of production and sales from which the BCa confidence intervals on the variance ratio are constructed.

(iii) Converting Equation (3)

Equation (3) can be written as

y t = μ ^ ′ + θ ^ D T C 1 t + β ^ ′ t + γ ^ D T C 2 t + O T ″ t . (3)'

Therein, D T C j ( T B j + i ) = ∑ h = 1 i [ D T j i − ( h − 1 ) ] α ^ ( h − 1 ) for i = 1 , 2 , ⋯ , N − T B , and

D T C j t = 0 for t ≤ T B j , where N stands for the number of observations, and where j = 1, 2.

(i) Seasonal dummy variables

Equation (3) with the seasonal dummy variables is written as

^{5}L represents the lag operator.

y t = μ ^ + φ 1 d 1 + φ 2 d 2 + φ 3 d 3 + θ ^ D T 1 t + β ^ t + γ ^ D T 2 t + α ^ y t − 1 + ∑ i = 1 k β ^ i Δ y t − i + e t . (4)

That equation can be converted to

y t = μ ^ ′ + φ 1 d c 1 + φ 2 d c 2 + φ 3 d c 3 + θ ^ D T C 1 t + β ^ ′ t + γ ^ D T C 2 t + O T ″ t . (4)'

1^{st} quarter | |||

2^{nd} quarter | |||

3^{rd} quarter | |||

4^{th} quarter |

Notes: The equation to test for a unit root of variable y t is y t = μ ^ + β ^ 1 d 1 t + β ^ 2 d 2 t + β ^ 3 d 3 t + γ ^ t + α ^ y t − 1 + e t , where di_{t} are the seasonal dummy variables that take a value of one in the ith quarter (i = 1, 2, and 3), and t, e t respectively denote the linear time trend and the residual. μ ^ , β ^ 1 , β ^ 2 , β ^ 3 , γ ^ , and α ^ are the parameters in the equation. The table presents values that the term d i t 1 − α ^ L = d c i t (I = 1, 2, 3) takes in each quarter (i = 1, 2, and 3) during observations.

in this equation is given as V ( y t ) = 1 4 ∑ i = 0 3 ( φ i − φ ¯ ) 2 + V ( O T ″ t ) as described in earlier reports [

the variance of O T ″ t .

Equations (1)', (2)', (3)', and (4)', or an appropriate combination of these equations, were applied to the production and the sales series. Bootstrap resamplings were conducted over the deterministic parts of the converted equations simultaneously, to construct the BCa confidence intervals for the relative variance ratio of production and sales.

One report of the relevant literature [

Japan | US | France | UK | |||||
---|---|---|---|---|---|---|---|---|

GDP | Sales | GDP | Sales | GDP | Sales | GDP | Sales | |

Obs. | (1955.1-1999.1) | (1967.1-2008.1) | (1978.1-008.1) | (1983.1-2008.1) | ||||

Period I | ||||||||

(1967.1-1984.4) | ||||||||

No. of Breaks | 2 | 2 | 2 | 2 | 0 | 1 | 1 | 1 |

Period II | ||||||||

(1985.1-2008.1) | ||||||||

No.of Breaks | 2 | 2 | ||||||

Period I | ||||||||

Break Type | BB | BB | AA | AA | N/A | C | A | A |

Period II | ||||||||

BB | BB | |||||||

Period I | ||||||||

TB1 | 1972.3 | 1972.3 | 1978.1 | 1978.1 | 1995.2 | 1990.1 | 1990.4 | |

TB2 | 1992.3 | 1992.3 | 1981.3 | 1981.3 | ||||

Period II | ||||||||

TB1 | 1995.2 | 1995.2 | ||||||

TB2 | 2000.2 | 2000.2 | ||||||

Period I | ||||||||

Level of Significance( ) | ** | ** | ** | * | * | * | ** | ** |

Period II | ||||||||

* | *** | |||||||

Data type | NSA | NSA | SA | SA | SA | SA | NSA | NSA |

Notes: Obs. and No. respectively stand for the sample period and number. τ denotes the t ratio of α ^ . *, **, and *** respectively denote 10%, 5%, and 1% levels of significance. NSA and SA respectively stand for not seasonally adjusted and seasonally adjusted series for quarterly data. TBi stands for the ith break point (i = 1, 2).

We divided the US sample period into two, and applied these periods to the models to test for a unit root. In each time period, the US data were trend stationary with two structural breaks. The confidence intervals for the variance ratio were therefore constructed in each time period for the US. The results for Germany in

Italy | Germany | Canada | ||||
---|---|---|---|---|---|---|

GDP | Sales | ΔGDP | ΔSales | GDP | Sales | |

Obs. | (1981.1-2007.1) | (1991.1-2008.1) | (1990.2-2008.1) | |||

No. of Breaks | 2 | 2 | 0 | 0 | 2 | 2 |

Break type | BC | BB | N/A | N/A | AB | CB |

TB1 | 1991.3 | 1991.3 | 1998.2 | 1995.4 | ||

TB2 | 2003.1 | 2003.3 | 2007.3 | 2002.1 | ||

Level of Significance (τ) | ** | ** | *** | *** | *** | *** |

Data type | NSA | NSA | NSA | NSA | SA | SA |

Germany | Canada | Canada | ||||
---|---|---|---|---|---|---|

ΔGDP | Sales | GDP | Sales | GDP | Sales | |

Obs. | (1957-1991) | (1965.1-1978.4) | (1992.1-2008.1) | |||

No. of Breaks | 0 | 0 | 0 | 0 | 1 | 1 |

Break type | N/A | N/A | N/A | N/A | B | B |

TB1 | 2003.4 | 2004.2 | ||||

TB2 | ||||||

Level of Significance (τ) | *** | *** | * | * | ** | ** |

Data type | Annual | Annual | SA | SA | SA | SA |

Notes: Obs. and No. respectively stand for the sample period and number. τ denotes the t ratio of α ^ . *, **, and *** respectively denote 10%, 5%, and 1% levels of significance. NSA and SA respectively stand for not seasonally adjusted and seasonally adjusted series for quarterly data. TBi stands for the i th break point (i = 1, 2).

Tables 4-12 present the BCa bootstrap confidence intervals of the 90%, 95%, and 99% confidence levels for G7 countries. In these tables, V(y)/V(s), z ^ 0 and a ^ respectively represent the ratio of the variance of production to the variance of sales, the bias correction constant, and the acceleration constant. θ ^ is the point estimate of the sample variance ratio. The number of bootstrap resample- ings is 3,000.

Japan, the US, France, and Italy show the production counter-smoothing property of inventory investment: The widest 99% confidence intervals represented are located in the region toward the right (greater) side of the value of one. Where the 90% confidence interval is applied, the UK is included in the group of countries for which production is more volatile than sales. Reports of the literature [

V(y)/V(s) | |
---|---|

Confidence Interval | BCa |

99% | (1.01335, 1.02716) |

95% | (1.01473, 1.02552) |

90% | (1.01555, 1.02448) |

Notes: z ^ 0 = − 0.00334 , a ^ = 0.04991 , and θ ^ = 1.01993 . The sample period for the unit root test is 1955.1-1999.1 (no. observations = 177). The observation periods used to construct BCa confidence intervals were set as 1958.2-1999.1 (no. observations = 164).

V(y)/V(s) | ||
---|---|---|

Confidence Interval | BCa | |

Period | I | II |

99% | (1.18870, 1.68141) | (1.1120, 1.5940) |

95% | (1.25243, 1.60796) | (1.16848, 1.52188) |

90% | (1.28194, 1.57434) | (1.19710, 1.48385) |

Notes: z ^ 0 = − 0.008356 , a ^ = − 0.002183 , and θ ^ = 1.420776 for period I, and z ^ 0 = − 0.29324 , a ^ = − 0.008192 , and θ ^ = 1.3542 for period II. Periods I and II for the unit root tests were1967.1-1984.4 (no. observations = 72) and 1985.1-2008.1 (no. observations = 93). Observation periods used to construct BCa confidence intervals for periods I and II were set respectively as1970.2-1984.4 and 1988.2-2008.1 (no. observations = 59 and 80).

V(y)/V(s) | |
---|---|

Confidence Interval | BCa |

99% | (1.084788, 1.694079) |

95% | (1.144478, 1.597168) |

90% | (1.180879, 1.552616) |

Notes: z ^ 0 = − 0.1916709 , a ^ = 0.012346076 , and θ ^ = 1.364265449 . The sample period for the unit root test was 1978.1-2008.1 (no. observations = 121). The observation period used to construct BCa confidence intervals was set as 1981.1-2008.1 (no. observations = 109).

V(y)/V(s) | |
---|---|

Confidence Interval | BCa |

99% | (0.99879, 1.00923) |

95% | (0.99968, 1.00733) |

90% | (1.00011, 1.00647) |

Notes: z ^ 0 = 0.06522 , a ^ = 0.04700 , and θ ^ = 1.00261 . The sample period for the unit root test was 1983.1-2008.1 (no. observations = 101). The observation period used to construct BCa confidence intervals was set as 1986.2-2008.1 (no. observations = 88).

V(y)/V(s) | |
---|---|

Confidence Interval | BCa |

99% | (1.00023, 1.00395) |

95% | (1.00073, 1.00361) |

90% | (1.00097, 1.00344) |

Notes: z ^ 0 = 0.05517 , a ^ = − 0.00131 , and θ ^ = 1.00227 . The sample period for the unit root test was 1981.1-2007.1 (no. observations = 105). The observation period used to construct BCa confidence intervals was set as 1984.2-2007.1 (no. observations = 92).

V(y)/V(s) | |
---|---|

Confidence Interval | BCa |

99% | (0.179115, 0.289420) |

95% | (0.189857, 0.271006) |

90% | (0.195748, 0.262982) |

Notes: z ^ 0 = 0.021726 , a ^ = 0.018947 , and θ ^ = 0.226948 . The sample period for the unit root test was 1991.1-2008.1 (no. observations = 69). The observation period used to construct BCa confidence intervals was set as 1991.4-2008.1 (no. observations = 66).

V(y)/V(s) | |
---|---|

Confidence Interval | BCa |

99% | (1.006113, 1.001788) |

95% | (1.148905, 1.852629) |

90% | (1.233023, 1.766839) |

Notes: z ^ 0 = − 0.090361 , a ^ = − 0.065289 , and θ ^ = 1.522056 . The sample period for the unit root tests was 1957-1991 (no. observations = 35). The observation period used to construct BCa confidence intervals was set as 1960-1991 (no. observations = 32).

V(y)/V(s) | |
---|---|

Confidence Interval | BCa |

99% | (0.823454, 1.843992) |

95% | (0.910669, 1.681195) |

90% | (0.963495, 1.592324) |

Notes: z ^ 0 = − 0.010027 , a ^ = − 0.041922 , and θ ^ = 1.233036 . The sample period for the unit root test was 1965.1-1978.4 (no. observations = 56). The observation period used to construct BCa confidence intervals was identical to the sample period.

V(y)/V(s) | |
---|---|

Confidence Interval | BCa |

99% | (0.578803, 1.333072) |

95% | (0.666948, 1.267492) |

90% | (0.716487, 1.204174) |

Notes: z ^ 0 = 0.274110 , a ^ = 0.002397 , and θ ^ = 0.916256 . The sample period for the unit root test was set as 1992.1-2008.1 (no. observations = 65). The observation period used to construct BCa confidence intervals was set as1995.2-2008.1 (no. observations = 52).

From

The probability that the calculation of the BCa bootstrap confidence intervals stops in the middle of the prescribed number of repetitions is not negligible when a break point of the time trend is located near the end of the sample period because the resampling process might then involve picking up only zeros for the dummy variable that represents a structural break. The second break point of the Canadian production series was found to be in the third period from the end of the sample period (

Point estimates of the relative variances of production and sales were used as the bases of the analyses in earlier reports of the relevant literature [

The confidence interval of the variance ratio enables interpretation of the test results of the production smoothing hypothesis of inventory investment, in terms of the relative volatility of production and sales in light of statistical variations of the sample data. A stylized fact [

Unlike results from the literature, results for Canada are indeterminate throughout the two time periods analyzed here. West German data show the production counter-smoothing property described in the literature. Germany since reunification, however, has less volatile production than sales.

It must be investigated whether the alleged international duality in the smoothing property of inventory investment remains as a remarkable feature in terms of the confidence interval estimations. Therefore, the analytical method presented in this paper should be implemented for other countries, particularly a group of economically developing countries.

When the sample period of a time series is sufficiently long to the degree that the number of break points exceeds two, then the number of combinations for the possible break types can render the computation burden implausible. The way to apply the methodologies represented in this paper in such occasions is to divide the sample period, so that each period constitutes a tractable dataset. This procedure, however, points to the shortcomings of the methods described in this paper in achieving consistent statistical analyses of the entire sample period of the data with a long time horizon. Another implicit disadvantage for the methods used for this study arises when a break point is located near the end of a sample period. Such was the case for the Canadian production time series in

Previous versions of this paper were presented at the Far East and South Asian Meeting of the Econometric Society (August 2009), and at the Singapore Economic Review Conference 2013 (August 2013). We are grateful to participants of those conferences for their helpful comments and discussions. Any remaining errors, however, are solely attributable to the authors.

The authors acknowledge financial support from the Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (KAKENHI) (Grant Number 23530221).

Ginama, I. and Odaki, M. (2017) Production Smoothing in Developed Countries. Journal of Mathematical Finance, 7, 333-350. https://doi.org/10.4236/jmf.2017.72018

Simply incorporating the ordinary dummy variables into Model (A) of [

y t = μ 1 + ( μ 2 − μ 1 ) D U t + θ t + ∑ i = 1 3 γ i d i t + α y t − 1 + ϵ t , ∀ t ≥ 1 , (5-1)

where D U t follows the definition presented in an earlier report [

Δ y t = μ ^ 1 + ( μ ^ 2 − μ ^ 1 ) D U t + θ ^ t + ∑ i = 1 3 γ ^ i d i t + ( α ^ − 1 ) y t − 1 + e ^ t , t = 2 , ⋯ , T , (5-2)

to obtain the test statistic, and in connection with Equation (5-2), let l T , l T B , τ T and l i ; T , i = 1, 2, 3 be the (T-1)-dimensional vectors composed respectively of 1, D U t , t and d i t , as

l T = ( 1 , ⋯ , 1 ) ′ , τ T = ( 2 , 3 , ⋯ , T ) ′ , l T B = ( 0 , ⋯ , 0 , 1 , ⋯ , 1 ) ′

provided that the ( T B + 1 ) th component is the first being 1,

l 1 ; T = ( 1 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , ⋯ ) ′ ,

l 2 ; T = ( 0 , 1 , 0 , 0 , 0 , 1 , , 0 , 0 , 0 , ⋯ ) ′ ,

l 3 ; T = ( 0 , 0 , 1 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , ⋯ ) ′ ,

with breakpoint T B defined in the described above articles. Furthermore, let λ = lim T → ∞ T B / T and further define

Y = ( y 2 , ⋯ , y T ) ′ , Y − 1 = ( y 1 , ⋯ , y T − 1 ) ′ , E ¯ = ( ϵ 2 , ⋯ , ϵ T ) ′ ,

S ¯ = ( ∑ h = 1 2 ϵ h , ∑ h = 1 3 ϵ h , ⋯ , ∑ h = 1 T ϵ h ) ′ , S ¯ − 1 = ( ϵ 1 , ∑ h = 1 2 ϵ h , ⋯ , ∑ h = 1 T − 1 ϵ h ) ′ ,

X 1 = [ l T , l T B , τ T ] , X 2 = [ l 1 ; T , l 2 ; T , l 3 ; T ] , X = [ X 1 , X 2 ] , Z 1 = [ Y − 1 , X 1 ] , Z = [ Y − 1 , X ] ,

M 1 T = I T − 1 − l T ( l ′ T l T ) − 1 l ′ T , M X i = I T − 1 − X i ( X ′ i X i ) − 1 X ′ i , i = 1 , 2 ,

M X = I T − 1 − X ( X ′ X ) − 1 X ′ ,

M Z 1 = I T − 1 − Z 1 ( Z ′ 1 Z 1 ) − 1 Z ′ , M Z = I T − 1 − Z ( Z ′ Z ) − 1 Z ′ ,

D T ; X = diag { T 1 / 2 , T 1 / 2 , T 3 / 2 , T 1 / 2 , T 1 / 2 , T 1 / 2 } , D T ; Z = [ T 0 0 D T ; X ] .

Letting T 4 be the largest integer that is either equal to or less than ( T − 1 ) / 4 and noting that

∑ t = 1 T 4 { i + 4 ( t − 1 ) } = i T 4 + 4 ∑ t = 1 T 4 − 1 t = i T 4 + 2 ( T 4 − 1 ) ( T 4 − 2 ) = 2 T 4 2 + O ( T ) = T 2 / 8 + O ( T ) , i = 1 , 2 , 3 ,

then it is not difficult to confirm that

D T ; X − 1 X ′ X D T ; X − 1 = [ 1 1 − λ 1 2 1 4 1 4 1 4 1 − λ 1 − λ 1 − λ 2 1 − λ 4 1 − λ 4 1 − λ 4 1 2 1 − λ 2 1 3 1 8 1 8 1 8 1 4 1 − λ 4 1 8 1 4 0 0 1 4 1 − λ 4 1 8 0 1 4 0 1 4 1 − λ 4 1 8 0 0 1 4 ] + O ( T − 1 ) (5-3)

The test statistic t α (i.e., the t-statistic of the regression coefficient ( α ^ − 1 ) in (5-2)) based on (5-2) is now expressed as

t α ( λ ) = ( T − 1 Y ′ M Z Y ) − 1 2 ( Y ′ − 1 M X Y − 1 ) − 1 2 { Y ′ − 1 M X ( Y − Y − 1 ) } .

Furthermore, let W A ( λ , r ) denote a stochastic process on [0, 1], which is the projection residuals in L 2 [ 0 , 1 ] of a Brownian motion projected onto the subspace generated by 1, du (λ, r), r with du (λ, r) such that du (λ, r) = 1 if r ≥ λ and 0 otherwise, as done by Equations (9) and (10) in the literature [

Lemma 1: Presume that y t is generated by (5-1). If the null is formulated as μ 1 = μ 2 , θ = γ 1 = γ 2 = γ 3 = 0 and α = 1 , and (5-2) is the regression for the unit root test, then the limiting distribution of t α ( λ ) under the null is expressed as

( ∫ 0 1 W A ( λ , r ) 2 d r ) − 1 / 2 ( ∫ 0 1 W A ( λ , r ) d W A ( λ , r ) ) .

Proof: First, for any integers i, j such that 4 > j > i > 0 and s = 1 , ⋯ , T 4 , ∑ n = i + 1 j ϵ 4 ( s − 1 ) + n is the sum of ( j − i ) pieces of ϵ 4 ( s − 1 ) + n and that ∑ n = i + 1 j ϵ 4 ( s − 1 ) + n and ∑ n = i + 1 j ϵ 4 ( s ′ − 1 ) + n are independent as s ≠ s ′ . It is then shown by these matters that for i, j given above

T − 3 / 2 ∑ s = 1 T 4 ∑ h = 1 4 ( s − 1 ) + j ϵ h − T − 3 2 ∑ s = 1 T 4 ∑ h = 1 4 ( s − 1 ) + i ϵ h = T − 3 / 2 ∑ s = 1 T 4 ( ∑ n = i + 1 j ϵ 4 ( s − 1 ) + n ) = O P ( T − 1 ) ,

which implies that

T − 3 / 2 ∑ s = 1 T 4 ∑ h = 1 4 ( s − 1 ) + i ϵ h = ( 1 4 ) T − 3 / 2 ∑ t = 2 T ∑ h = 1 t − 1 ϵ h + O P ( T − 1 ) , i = 1 , 2 , 3 , (A.1)

noting that

T − 3 / 2 ∑ t = 2 T ∑ h = 1 t − 1 ϵ h = ∑ i = 1 3 T − 3 / 2 ∑ s = 1 T 4 ∑ h = 1 4 ( s − 1 ) + i ϵ h .

Next, note that the ( T − 1 ) observations of Equation (5-1) under the null are written as

Y = Y − 1 + E ¯ + μ 1 l T = Y − 1 + X b + E ¯ = Z ( 1 , b ′ ) ′ + E ¯ ,

where b is the coefficient vector defined suitably. This expression engenders

t α ( λ ) = ( T − 1 E ¯ ′ M Z E ¯ ) − 1 / 2 ( T − 2 Y ′ − 1 M X Y − 1 ) − 1 2 ( T − 1 Y ′ − 1 M X E ¯ ) , (A.2)

noting that

T − 2 Y ′ − 1 M X Y − 1 = T − 2 Y ′ − 1 M X 1 Y − 1 − ( T − 3 2 Y ′ − 1 M X 1 X 2 ) ( T − 1 X ′ 2 M X 1 X 2 ) − 1 ( T − 3 2 X ′ 2 M X 1 Y − 1 ) , (A.3)

T − 1 Y ′ − 1 M X E ¯ = T − 1 Y ′ − 1 M X 1 E ¯ − ( T − 3 2 Y ′ − 1 M X 1 X 2 ) ( T − 1 X ′ 2 M X 1 X 2 ) − 1 ( T − 1 2 X ′ 2 M X 1 E ¯ ) .

In view of Equation (5-3) and

y t − 1 = ∑ h = 1 t − 1 ϵ h + ( t − 1 ) μ 1 + y 0 ,

because of Equation (5-1) under the null and noting that

M X 1 = M 1 T − M 1 T τ ( τ ′ M 1 T τ ) − 1 τ ′ M 1 T ,

it is trivial to see that

( T − 1 X ′ 2 M X 1 X 2 ) − 1 = O ( 1 ) , (A.4)

and that

T − 3 2 Y ′ − 1 M X 1 X 2 = T − 3 2 S ¯ ′ − 1 M X 1 X 2 = T − 3 2 ∑ t = 2 T s ¯ t − 1 ( d 1 t , d 2 t , d 3 t ) − ( T − 3 2 ∑ t = 2 T s ¯ t − 1 ) { ( T − 1 ) − 1 ∑ t = 2 T ( d 1 t , d 2 t , d 3 t ) } − ( T − 5 2 S ¯ ′ − 1 M 1 T τ T ) ( T − 3 τ ′ T M 1 T τ T ) − 1 [ T − 2 ∑ t = 2 T t ( d 1 t , d 2 t , d 3 t ) − T − 2 ( ∑ t = 2 T t ) { ( T − 1 ) − 1 ∑ t = 2 T ( d 1 t , d 2 t , d 3 t ) } ] , (A.5)

where s ¯ t − 1 = ∑ h = 1 t − 1 ϵ t . Because T 4 T = 1 4 + O ( T − 1 ) and d i t = 1 for any t expressed

as t = 4s + i with a nonnegative integer s, 0 otherwise, by definition, we obtain

T − 3 / 2 ∑ t = 2 T s ¯ t − 1 ( d 1 t , d 2 t , d 3 t ) = T − 3 / 2 ∑ s = 1 T 4 ( ∑ h = 1 4 ( s − 1 ) + 1 ϵ h , ∑ h = 1 4 ( s − 1 ) + 2 ϵ h , ∑ h = 1 4 ( s − 1 ) + 3 ϵ h ) + O P ( T − 1 ) ,

( T − 3 / 2 ∑ t = 1 T s ¯ t − 1 ) { ( T − 1 ) − 1 ∑ t = 2 T ( d 1 t , d 2 t , d 3 t ) } = ( T − 3 / 2 ∑ t = 2 T ∑ h = 1 t − 1 ϵ h ) ( 1 4 , 1 4 , 1 4 ) + O P ( T − 1 ) ,

T − 2 ∑ t = 2 T t ( d 1 t , d 2 t , d 3 t ) = ( 1 8 , 1 8 , 1 8 ) + O ( T − 1 )

T − 2 ( ∑ t = 2 T t ) { ( T − 1 ) − 1 ∑ t = 2 T ( d 1 t , d 2 t , d 3 t ) } = ( 1 8 , 1 8 , 1 8 ) + O ( T − 1 ) . (A.6)

Using Equation (A.6) as well as Equation (A.1), (A.5) is converted into

T − 3 / 2 Y ′ − 1 M X 1 X 2 = O P ( T − 1 ) . (A.7)

It is then derived from (A.3), (A.4), and (A.7) that

T − 2 Y ′ − 1 M X Y − 1 = T − 2 Y ′ − 1 M X 1 Y − 1 + O P ( T − 2 ) ,

T − 1 Y ′ − 1 M X E ¯ = T − 1 Y ′ − 1 M X 1 E ¯ + O P ( T − 1 ) . (A.8)

It also follows from the well-known asymptotic results on the I(0) and I(1) series that

T − 1 E ¯ ′ M Z E ¯ = T − 1 E ¯ ′ E ¯ + O P ( T − 1 ) = σ 2 + O P ( T − 1 2 ) ,

T − 1 E ¯ ′ M Z 1 E ¯ = T − 1 E ¯ ′ E ¯ + O P ( T − 1 ) = σ 2 + O P ( T − 1 2 ) . (A.9)

Putting Equations (A.2), (A.8), and (A.9) together shows that the limiting distribution of t α ( λ ) is equal to that of

( T − 1 E ¯ ′ M Z 1 E ¯ ) − 1 2 ( T − 2 Y ′ − 1 M X 1 Y − 1 ) − 1 2 ( T − 1 Y ′ − 1 M X 1 E ¯ ) .

One readily finds that this quantity is the same as that derived from the test statistic for Model (A) of earlier papers [

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