Test for Unit Root

Looking at (3.11), one concern may arise about the stationarity of the data: not only the level of the time series but also the growth rate of the industry time series could display persistent effects of shocks and non-stationarity behavior. However, containing the EU KLEMS dataset time series with a maximum of 35 yearly observations per industry, it is quite difficult to detect non-stationarity with just one procedure. Therefore, a set of tests and checks should be performed.

At the first stage, it could be useful to plot the variable to check that some linear trends are present in the time series. Figure 3.4 contains three plots respectively for value added, investment and employment annual growth rate for the total economy.

The figure shows no observable trend in data, which suggests that a linear trend should be considered in the test. Similar behavior can be observed for the industry time series.

The first tests I can perform are those introduced by Dickey and Fuller (1979) in the augmented version (ADF) and Phillips and Perron (1988) (PP). Given a time seriesX_t, the ADF considers the following AR(n) model:

∆X_t=γ₀+φXt−1−

n

X

j=1

γ_j∆Xt−j+_t (3.52)

and performs the test for a unit root is then carried out under the null hypothesis ˆφ= 0 against the alternative hypothesis of φ < 0. Alternatively, the PP test considers a reduced form of (3.52), which is the original Dickey-Fuller test, i.e.,

∆Xt=γ0+φXt−1+t, (3.53)

and test the parameterφconsidering a non-parametric correction to the t-test statistic.

Table 3.5 provides the results for the ADF and PP tests without a trend and 2 lags for each sector and the total economy.⁷A first look at the Phillips Perron test rejects the null hypothesis of a unit root for all of the analyzed series. The null hypothesis is not rejected for some series (including all the time series for textiles and the output growth for agriculture).

7I also try to consider different values of the lags, especially, following Schwert (2002),inth

12T 100 1 4i

with T number of observations. The results are always consistent.

3.7. Empirical Application: Danish KLEMS Dataset

Table 3.4.: Danish industries and relationship-specificity

Industry Most

relationship-specific industry

AGRICULTURE FOOD

MINING AND QUARRYING ELECTRICITY

FOOD , BEVERAGES AND TOBACCO AGRICULTURE

TEXTILES, TEXTILE , LEATHER AND FOOTWEAR TRANSPORT

WOOD AND OF WOOD AND CORK ELECTRICITY

PULP, PAPER, PAPER , PRINTING AND PUBLISHING WHOLESALE

Chemicals and chemical AGRICULTURE

Rubber and plastics AGRICULTURE

OTHER NON-METALLIC MINERAL TRANSPORT

MACHINERY, NEC TRANSPORT

ELECTRICAL AND OPTICAL EQUIPMENT ELECTRICITY

TRANSPORT EQUIPMENT ELECTRICTITY

ELECTRICITY, GAS AND WATER SUPPLY ELECTRICITY

CONSTRUCTION Wholesale

Sale Wholesale

Wholesale Retail

Retail trade Wholesale

HOTELS AND RESTAURANTS WHOLESALE

POST AND TELECOMMUNICATIONS ELECTRICTY

Figure 3.4.: Value added, Investment and Employment Growth Rate. Total Economy.

3.7. Empirical Application: Danish KLEMS Dataset

Table3.5.:ResultsfortheADFandPPtestswithnotrendandwith2lags. IndustryAugmentedDickeyFullerPhillipsPerron Growthrate:OutputInvestmentEmploymentOutputInvestmentEmployment TotalIndustries-3.663-3.055-3.014-6.497-24.762-19.372 AgricultureandFishing-4.355***-3.724-2.829-11.519-24.408-27.632 Miningandquarrying-4.298***-3.042-4.451-25.280-31.521-25.346 Food,BeveragesandTobacco-2.002-3.722-3.332-52.356-42.534-22.087 Te*tiles-2.279**-2.556***-2.341***-33.458-41.840-37.435 Wood-3.364*-5.634-2.339***-38.371-36.795-22.041 Paperandprinting-3.744-2.952*-3.198-30.439-41.947-34.478 Coke-2.572***-3.574-3.256-39.219-38.784-38.323 Chemicals-3.781-3.185*-3.947-34.696-43.279-27.334 Rubber-3.902-4.011-3.218-27.657-32.866-18.079 Othernonmetal-3.323-3.445-3.697*-34.145-30.225-25.259 Basicmetals-3.736-4.420-4.490-28.546-24.216-21.754 Machinery,NEC-4.074-3.024-3.941-31.890-26.631-28.774 Electricalandopticalequip.-3.242-2.302*-3.407*-33.996-37.974-17.616 TransportEquipment-3.564*-3.789-3.213*-33.963-32.891-17.777 Recycling-3.697*-3.634*-2.508***-33.120-40.336***-16.261 Electricity,gasandwater-3.795-2.559***-3.233*-42.089-24.227***-23.333 Construction-3.502*-4.054-3.636-30.599-29.388-22.102 Sale-2.799**-3.186-2.580*-33.675-26.183-25.160 Wholesale-3.681*-2.802-2.272*-33.981-28.782-34.893 Retail-2.749*-2.919*-2.044*-32.625-35.148-17.055 Hotels-3.462*-4.187-3.396*-34.116-28.540-19.733 Transport-4.235-3.675*-0.529***-32.519-37.349-24.296 ***=significantatthe0.1%level. **=significantatthe1%level. ***=significantatthe5%level.

Table 3.6.: Panel unit root analysis: IPS test.

Growth: t-bar

Output -3.455

Investment -4.050

Employment -3.331

IPS test assumes the null hypothesis of stationarity.

Critical values for rejection:

1%: -1.7550 5%: -1.810 10%: -1.930

However, in this case, the test for short time series has relatively low power. To be sure that the unit root does not affect the time series, we consider the test developed by Im et al. (2003) (IPS), which is useful for testing panel unit roots. The IPS test can be defined as a cross-sectional average of the ADF test for the individual sectoral equation.

Table 3.6 shows that the hypothesis of non-stationarity can be strongly rejected.

I now apply the results obtained by the Kalman filter assuming that the initial con-dition is obtained by the Malmquist index procedure obtained from the procedures explained in Section 3.3 using both (3.32) and (3.40). I compare these results with the Solow-Törnqvist Index obtained from the EU KLEMS statistics (variableV AconT F P of the dataset). These Solow residuals are computed as the difference between the growth rate of the value added and the growth rate of the inputs, while the factor shares are the two period average shares of the input in nominal value added assuming constant returns to scale. As an example of the procedure, Figure 3.6 shows a good approximation of the Malmquist index if it is compared with the results obtained by the Gibbs-sampler, while Figure 3.5 shows the differences between the Kalman filter and the Solow residual, respectively, and the initial condition.

Tables 3.7 and 3.8, considering respectively (3.39) and (3.40), display the comparison between these methodologies over the entire period, 1970-2005, and two sub-periods, 1985-1994 and 1995-2005. I consider 1995 as an important break year for the initiation of massive investment in ICT in the US (Stiroh (2002b)) and in Europe (Dahl et al. (2009)).

Moreover, the upper part of these Tables represents the comparison for industries with a better-measured input.

Very similar patterns can be found in Table 3.8, where the coefficients of variation are almost identical (about 0.7). What is striking is that, while the Kalman Filter results are almost identical when comparing the two different Malmquist indexes used for the initial conditions, it is possible to find a difference for the less measurable industries.

For example, the coefficients of variation of the Solow residual for the less measurable industries are 0.8 and 2.3 for the periods 1985-1994 and 1995-2004, respectively, while the same statics for the Kalman filter are lower (0.6 and 0.7, for the periods 1985-1994 and 1995-2004, respectively). These results suggest that capital could still be measured with some bias in the statistics especially for the less measurable industries. Moreover, this problem is accentuated after during theNew Economy period.

3.7. Empirical Application: Danish KLEMS Dataset

Table3.7.:TFPMeasurementinDanishIndustries:AComparison(firsttable) Industry1970-20051985-19941995-2004 EUKLEMSK.F.EUKLEMSK.F.EUKLEMSK.F. Measurableindustries AgricultureandHunting6.35.711.46.01.75.7 Mining4.33.32.43.5-2.23.3 Machinery3.73.43.03.3-0.13.6 ManufacturingNEC0.10.12.00.0-1.00.1 Electricity,GasandWater1.30.9-2.71.00.01.0 Wholesale1.61.92.91.81.81.8 HotelsandRestaurants-1.6-1.5-5.0-1.6-3.2-1.6 Lessmeasurableindustries Food,BeveragesandTobacco1.82.40.42.8-1.61.9 Textiles2.41.3-1.81.5-0.21.3 Wood0.70.5-4.90.4-0.70.9 Pulp,PrintingandPublishing0.60.2-5.60.30.30.3 Chemicals4.83.25.93.11.82.9 RubberandPlastics1.91.6-3.21.5-1.41.6 OtherMinerals0.50.50.20.3-1.90.6 TransportEquipment0.70.22.1-0.1-1.90.9 SaleMotorVeichles-0.10.13.2-0.2-0.80.0 Retail0.50.62.90.6-0.20.8 PostandTLC1.81.95.82.13.51.7 EUKLEMSdenotesSolowresidual K.F.denotestheTFPgrowthestimatedfromtheState-spacemodel

Table3.8.:TFPMeasurementinDanishindustries:AComparison(secondtable) Industry1970-20051985-19941995-2004 EUKLEMSK.F.EUKLEMSK.F.EUKLEMSK.F. Measurableindustries AgricultureandHunting6.36.011.45.91.76.2 Mining4.33.42.43.3-2.23.5 Machinery3.73.43.03.3-0.13.5 ManufacturingNEC0.10.02.00.1-1.0-0.1 Electricity,GasandWater1.30.9-2.70.80.01.0 Food,BeveragesandTobacco1.82.30.42.3-1.62.3 Wholesale1.61.92.92.11.81.5 HotelsandRestaurants-1.6-1.4-5.0-1.5-3.2-1.2 Lessmeasurableindustries Textiles2.41.5-1.81.5-0.21.4 Wood0.70.5-4.90.4-0.70.5 Pulp,PrintingandPublishing0.60.1-5.6-0.10.30.3 Chemicals4.83.55.93.51.83.5 RubberandPlastics1.91.3-3.21.3-1.41.2 OtherMinerals0.50.30.20.3-1.90.2 TransportEquipment0.7-0.22.1-0.3-1.9-0.1 SaleMotorVeichles-0.1-0.13.2-0.0-0.8-0.2 Retail0.50.52.90.3-0.20.9 PostandTLC1.81.65.81.63.51.5 EUKLEMSdenotesSolowresidual K.F.denotestheTFPgrowthestimatedfromtheState-spacemodel

3.8. Conclusion

Finally, in Figure 3.6 the Törnqvist Index (green line) is compared to the Kalman Filter estimation (blue line).

3.8. Conclusion

Different econometric techniques of estimation and computation of technological change have been analyzed in the literature; however, when inputs, especially capital, are af-fected by measurement errors, the results may be biased. In this chapter, I propose a new methodology based on the State-space model and adopting Kalman Filter techniques, through which it is possible to estimate TFP growth without considering investment series instead of capital. I also analyze the problem for the initial condition of TFP growth. I compare four different approaches: the Solow residual, the Maximum Likeli-hood estimation, the Gibbs-sampler and the Malmquist Index. Comparing the results using the Gibbs-sampler, it is possible for the root mean squared of this procedure to be two-thirds lower than the Solow residual when capital contains measurement error.

In addition, I extend this framework to panel data. The empirical application utilizes Danish industry data. The comparison between the TFP growth measures provided by the Danish national statistics and the Kalman filter estimations suggests that capital can play an important in estimating technological change, especially for industries where it is more difficult to measure the inputs precisely.

Figure 3.5.: Initial value for the Danish industry: Malmquist index procedure and Gibbs-Sampler

3.8. Conclusion

Figure 3.6.: Danish retail: Törnqvist index and Kalman filter estimation