Saturday, August 18, 2007
I have also realised that I had slightly misinterpreted the methodology for calculating the AES. Whilst originally i was under the impression that I will need to calculate each AES for each data point for some reason (which i can't recall) I changed this interpretation to calculate the AES for each industry only. Hence, I was substituting in industry averages the AES formula (see previous post) as opposed to the values for each data point. Fortunately, this was realised quite early on and given that I have a more reliable data set I have not wasted too much time.
In my previous post I mentioned that I will be using a combination of R, Excel and Maxima to calculate the various AES's. I have now revised this to using Maxima to only simplify the differential equations to be used in the bordered Hessian matrix and use R solely for my regression analysis and AES calculations. My earlier post mentioned that R had a function via the micEcon package that calculates the bordered Hessian matrix. However since my translog model takes into account time and industry effects I will most likely need to write my own function. This should not pose too difficult though the somewhat low level nature of R can be quite frustrating especially when you are used to writing code in high-level OO languages. One of the most frustrating aspects of R is that not only does it not have inbuilt functions to determine the minor or cofactor matrix but there are no nice matrix operation like in Matlab where you can easily element rows and columns of a matrix to create a new matrix. After a few hours of searching I had to accept the fact that i needed to write my own cofactor function as well as copy the elements from the matrix whose cofactor we wish to determine element by element to create the matrix minor to be used in cofactor calculations. The function that I have written can be viewed below:
At present I am very busy with my analysis and completing my partial treatise draft and my next post will be later on next week after my meeting next Friday with Dr. Poon.
Saturday, August 11, 2007
Model Construction and Data
- SK: Software Capital
- HK: Hardware Capital (includes computer and peripheral but machinery and other electrical equipment)
- OK: Organisation Capital (Difference between Capital Stock and Software Capital and Hardware Capital)
- LH: Labour Hours
- VA: Value Added
The productivity function that I applied is the translog-production function similar to Hitt (2002), model (1). Click on the image to enlarge and view the equation.
The choice for this model was that it does not constrain the elasticities to be unary.
Since I am assuming that the model is a Classical Linear Regression Model (CLRM) and applying Ordinary Least Squares (OLS) to estimate the parameters of the model I need to stack the data in such a way that OLS could be applied. To achieve this I stacked the data over time for each industry as per below.
I then transformed the data to be in accordance with the model (1) and a pooled OLS was then run in R to find the coefficients. The results of the regression are below:
lm(formula = lnVa ~ lnSk + lnHk + lnOk + lnLh + lnSklnHk + lnSklnOk + lnSklnLh + lnHklnOk + lnHklnLh + lnOklnLh + lnSk2 + lnHk2 + lnOk2 + lnLh2)
Min 1Q Median 3Q Max
-0.58722 -0.16982 -0.02798 0.19408 0.72151
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.493241 6.164181 1.378 0.16979
lnSk -7.735495 1.051449 -7.357 4.69e-12 ***
lnHk 4.858182 0.820692 5.920 1.37e-08 ***
lnOk -3.002444 0.727414 -4.128 5.37e-05 ***
lnLh 4.214768 1.148203 3.671 0.00031 ***
lnSklnHk -0.008317 0.093900 -0.089 0.92951
lnSklnOk 0.586699 0.088497 6.630 3.03e-10 ***
lnSklnLh 0.064016 0.058670 1.091 0.27652
lnHklnOk -0.390088 0.063354 -6.157 3.96e-09 ***
lnHklnLh -0.032604 0.044211 -0.737 0.46171
lnOklnLh -0.416532 0.053405 -7.799 3.31e-13 ***
lnSk2 0.067969 0.054974 1.236 0.21775
lnHk2 -0.018647 0.042730 -0.436 0.66303
lnOk2 0.290936 0.044207 6.581 3.97e-10 ***
lnLh2 0.032540 0.039548 0.823 0.41159
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2555 on 201 degrees of freedom
Multiple R-Squared: 0.8219, Adjusted R-squared: 0.8095
F-statistic: 66.26 on 14 and 201 DF, p-value: <>
It is evident that some of the results are in contradiction to the underlying economic theory but this can be owing to the quality of the data (more about this later).
Calculation of AES
To calculate the AES I needed four items:
- Averages of the independent of variables for each industry
- Partial first order derivatives, second order derivatives of the model(1)
- Bordered Hessian matrix for each industry
- Appropriate co-factor between the independent variables for each industry
Below are the results of the partial AES between SK and the other independent variables for each industry. As mentioned earlier these are not highly accurate due to the problems with the underlying data. They are included to show that the calculation methodology employed is sound.
Next Weeks Deliverables
As I have mentioned throughout my post the results did contradict economic theory. Dr. Poon has suggests two things. Firstly, that I rerun my model to include a time effect variable which corresponds to every 5 years. The model will now become:
Secondly, that I worry about the results of the data later and that I should now concentrate on polishing up my calculation methodology to ensure that my procedures are correct and accurate. Dr.Poon has informed me that in my final treatise I will require a detailed description of the process that I undertook in these calculations. Furthermore Dr.Poon has also provided us with more details on the overall structure of our treatise. Since our partial treatise write up is due on August 31 Dr.Poon hopes that we will be able to complete, the Introduction & Motivation, Literature Review and Data set construction technique by then. Once I have finalised my calculation process I will publish it on the blog.
Monday, August 6, 2007
Today I met up with Gary to continue work on the construction of the data set. However we were still slightly confused as to how some of the items were calculated. Fortunately Dr. Poon was in his office when we were working in the near by lab and we were able to clarify this. Our major concern was that our reconstruction of the data set was not matching up exactly to the numbers that Dr.Poon had calculated for each variable that is:
- Productive Capital Stock
- Labour Hours
- Value Added
Dr. Poon reassured that the methods we were using were CORRECT. The differences in the numbers was due how the ABS calculates Chain Volume Measures (CVM). CVM were used for each of the variables collected. CVM are defined as an index number which allows the base year to be updated more regularly and thus the statistics will be more accurate. As the CVM’s that Dr.Poon used in his calculation for productive capital stock, labour hours and value added were different to the CVM’s calculate for these variables we were going to get different results for our calculations. As a result our data collection and construction task is greatly simplified.
The only difficult item that remains is to find out if there have been any changes in the deflators used as Dr. Poon would like us to use Current Prices as opposed to CVM. Nik will be contacting the ABS shortly to enquire if any changes have been made. We may also need to contact the ABS late on if we are going to determine Value-Add for the private and public sector. This is because the ABS reports the data as combined amount of both sectors. However, Dr. Poon informed me that this is not of high importance at present and just to ensure that we have the data set constructed for the two sectors combined.
Implementation and construction of the Allen Partial Elasticity of Substitution (AES)
In addition to gaining clarity on the finer details of the data set construction I was also able to clarify a few questions which I had about calculating AES. The first item which was clarified was the items of the border Hessian matrix. All I need to do is calculated the partial first order and second order derivatives and obtain a numerical for each element using the results obtained from the pooled regression and industry averages of the variables. Dr.Poon has instructed me to use the program Maxima ,which will greatly simplify the calculation of these derivatives. More information about maxima can be found at: http://maxima.sourceforge.net/. I will now spend some time in becoming more familiar with Maxima to calculate these derivatives.
I was also able to clarify the calculation of the standard error of the AES which will be used to see if it statistically significant. Previously I thought that I will need to calculate the standard error for each AES for each industry. However Dr.Poon informed me that I will just need to find the standard error for the each AES across ALL industries. This once again simplifies my task. I will shortly be posting up details of my results for the various AES’ that I will calculate.
Thursday, August 2, 2007
Data Set Construction
Dr.Poon provided further clarification as to the construction of the new data set. Most importantly I was able to attain a greater understanding as to how Dr. Poon constructed the productive capital component of his data set. He informed me that the two ABS items that were used in his data construction were:
- Net Capital Stock
- Productive Capital
The use of the appropriate deflators is another important issue. Sine we are looking at a panel data set (a set of variables across time and a range of industry) we will need to express our units in constant dollar terms. Dr. Poon informed us that we may need to ring the ABS to find out how the calculated it for particular items if it not available on the website. This will be further investigate by my fellow peers and I in the coming week. Dr. Poon has also asked us to express our data set using the following variables:
- Software Capital
- Hardware Capital
- Other Capital (essentially anything that is not classified as IT capital)
- Labour Hours
- Value Add (output)
In addition to the construction of the data set Dr. Poon set each of us a particular task involving the data set he provided. My task involves calculating the AES in the R using the technique i mentioned in my earlier post for the data set he has provided. I plan to go about this in the following way:
- Splice up the data set into the following variables: Software Capital, Hardware Capital, Other Capital, Labour Hours, Value Add, for each industry across time
- Stack the panel data set so a pooled regression can be run
- Impose a translog function, transform the data appropriately and run the regression
- Using the estimates of the parameters from the regression calculate the bordered Hessian matrix
- Calculate the AES for all input variables for all industries
- Determine the Standard error for each AES calculated
- Apply a T-test to each AES to see if it is statistically significant or not.