The data are the most comprehensive estimates we’ve done to date, covering 426 industries and the more than 5,300 products produced by them, ranging from soy beans to jet fuel refining. The data are formatted into two easy-to-use Excel files, one with the industry estimates and one for the product estimates. (See a preview of the data here.)

These data are a terrific short-cut for researchers, students and others looking to estimate the carbon impact of various products or industries without building their own input-output or CGE model to do so. All estimates are based on our Chamberlain Economics, L.L.C. Input-Output Model, which is designed to model carbon flows in the U.S. economy.

See our “Data Products” page for ordering information.

In this post I’ll explain how to solve a common problem we’ve run into: how to divide quarterly data into monthly data for econometric analysis. To do so, we’ll use a method known as “cubic spline interpolation.”

Cubic Spline Interpolation
One of the most widely used data sources in economics is the National Income and Product Accounts (NIPAs) from the U.S. Bureau of Economic Analysis. They’re the official source for U.S. GDP, personal income, trade flows and more. Unfortunately, most data are published only quarterly or annually. So if you’re hoping to run a regression using monthly observations — for example, this simple estimate of the price elasticity of demand for gasoline — you’ll need to split these quarterly data into monthly ones.

A common way to do this is by “cubic spline interpolation.” Here’s how it works. We start with n quarterly data points. That means we have n-1 spaces between them. Across each space, we draw a unique 3rd-degree (or “cubic”) polynomial connecting the two points. This is called a “piecewise polynomial” function.

To make sure our connecting lines form a smooth line, we force all our first and second derivatives to be continuous; that is, at each connecting point we make them equal to the derivitive on either side. When all these requirements are met — along with a couple end-point conditions you can read about here — we have a (4n-4) x (4n-4) linear system that can be solved for the coefficients of all n-1 cubic polynomials.

Once we have these n-1 piecewise polynomials, we can plug in x values for whatever time intervals we want: monthly, weekly or even daily. The polynomials will give us a pretty good interpolation between our known quarterly data points.

An Example Using MATLAB
While the above method seems simple, doing cubic splines by hand is not. A spline for just four data points requires setting up and solving a 12 x 12 linear system, then manually evaluating three different polynomials at the desired x values. That’s a lot of work. To get a sense of how hard this is, here’s an Excel file showing what’s involved in fitting a cubic spline to four data points by hand.

In practice, the best way to do a cubic spline is to use MATLAB. It takes about five minutes. Here’s how to do it.

MATLAB has a built-in “spline()” function that does the dirty work of cubic spline interpolation for you. It requires three inputs: a list of x values from the quarterly data you want to split; a list of y values from the quarterly data; and a list of x values for the monthly time intervals you want. The spline() function formulates the n-1 cubic polynomials, evaluates them at your desired x values, and gives you a list of interpolated monthly y values.

Here’s an Excel file showing how to use MATLAB to split quarterly data into monthly. In the file, the first two columns are quarterly values from BEA’s Personal Income series. Our goal is to convert these into monthly values. The next three columns (highlighted in yellow) are the three inputs MATLAB needs: the original quarterly x values (x); the original quarterly y values (y); and the desired monthly x values (xx).

In the Excel file, note that the first quarter is listed as month 2, the second quarter as month 5, and so on. Why is this? BEA’s quarterly data represent an average value over the three-month quarter. That means they should be treated as a mid-point of the quarter. For Q1 that’s month 2, for Q2 that’s month 5, and so on.

The next step is to open MATLAB and paste in these three columns of data. In MATLAB, type ” x = [ ", cut and paste the column of x values in from Excel, type " ] ” and hit return. This creates an n x 1 vector with the x values. Repeat this for the y, and xx values in the Excel file.

Once you have x, y, and xx defined in MATLAB, type “yy = spline(x,y,xx)” and hit return. This will create a new vector yy with the interpolated monthly y values we’re looking for. Each entry in yy will correspond to one of the x values you specified in the xx vector.

Copy these yy values from MATLAB, paste them into Excel, and we’re done. We now have an estimated monthly Personal Income series.

Here’s an Excel file summarizing the above example for splitting quarterly Personal Income data into monthly using MATLAB. Also, here’s a MATLAB file with the x, y, xx, and yy vectors from the above exercise.

The study is basically a critique of recent Congressional Budget Office (CBO) distributional estimates that suggest the bill’s impact is likely to be progressive across income groups. We find the bill is much more likely to be regressive across income groups once the microeconomic response of regulated public utilities is taken into account. Under this framework, we estimate the bill will result in net benefits to the nation’s highest-earning 20 percent of earners, while imposing net costs on the lowest-earning 80 percent of U.S. households.

Check out the full study and news release here.

You can view the full study here.

To help our clients better understand the economic theory underlying our work, we’ll be posting an ongoing series of articles titled “Core Concepts.” The goal is to provide a collection of simple and brief introductions to the core theoretical concepts used by Chamberlain Economics, L.L.C.

As the first in the series we’ve posted “Core Concepts: The Economics of Tax Incidence“. The piece is designed as a refresher on the basics of tax incidence, and how it’s derived analytically from elasticities of supply and demand in the marketplace. This idea serves as the foundation for nearly all of our work on tax modeling and policy analysis.

Check out the article here.

Instead of producing a square industry-by-industry input-output table, the Bureau of Economic Analysis (BEA) produces rectangular “use” and “make” tables. The make table shows products produced by each industry, while the use table shows how products get used by industries, consumers, government and the rest of the world. However, what we need for Leontief models is a square table that shows only the industry-by-industry relationships.

I’ve you’re a researcher who has run into this problem, I have good news. Our economists have produced summary-level (134 industries) and detailed-level (426 industries) input-output tables from BEA data, which are now for sale on our “Data Products” page. We’ve also written up a methodology paper explaining how the tables are derived, allowing you to reproduce them quickly and easily. See here to place an order today.

Gross receipts taxes like the B&O tax work like a sales tax, except they apply to business inputs as well as final goods. For a baker selling loaves of bread, the flour, electricity and packaging are all taxed first, then the loaf itself is taxed when sold to consumers. These extra layers of taxation get quietly built into the final selling price—something economists call “tax pyramiding.”

Here’s the abstract for the piece:

Using newly released 2002 Washington State input-output data, we provide the first estimates of tax pyramiding from the state’s Business & Occupation (B&O) tax since 2001. We find tax pyramiding is more severe than found by previous studies that did not distinguish between imported and domestically produced products. We find the B&O tax pyramids an average of 3.0 times, ranging from 1.6 times on architectural, engineering and computing services to 16.7 times on petroleum and coal products manufacturing.

A file with some tables of findings is here. If you’d like to learn how we can develop an input-output model like this for your own study, give us a call today.

Like most concepts in economics, price elasticity is easy to talk about but hard to measure.

In this post, I’ll show you how Chamberlain Economics measures demand elasticities in the real world. We’ll develop a simple theory, write it down mathematically, find some data and crunch the numbers in Excel. At the end, I’ll hand over a spreadsheet with our own elasticity estimates for retail gasoline that replicate the numbers from a well-known recent econometric study.

Start with Theory
Our goal is to estimate the price elasticity of demand for retail gasoline. The first step is to start with a theory about the demand for gas.

The simplest theory is that we know gasoline — like everything else — should have a demand curve. What should it look like? In the simplest case, it should be driven by two things: the price of gas, and how much income people have. If gas prices rise consumption should fall; conversely, if income goes up gas consumption should rise also.

So there’s our theory. Now, let’s write it down mathematically. If gas demand is a function of prices and income, one way we can write it is like this:

G = a*P + b*Y

Where:

G = Gallons of gas demanded per year
P = The price of gas
Y = Average income in the economy
a, b = Coefficients for the magnitude of the impact of prices and income on gas demand. (Note: According ot our theory, the “a” coefficient on prices should be a negative number and the “b” coefficient on income should be positive.)

Now that we’ve got a theory, the next step is to translate it into a form we can estimate in the real world. Think of the theory as an architect’s drawing — it’s a guide, but our goal is to actually to build it with hammer and nails.

To do this, think about real-world factors that might complicate our simple theory. For one, we should probably control for population by using per capita figures. Next, we should control for inflation by inflation-adjusting everything. Finally, we should control for seasonal variation somehow, since gas demand always peaks in summer and slows in winter.

Taking these messy details into account, here’s how we translate our theory into a relationship we can actually estimate. Economists call this “specifying the model”:

Gij = A + a*Pij + b*Yij + ei + eij

Where:

G = Per capita gas demand in month i and year j
A = The y-intercept term in our linear demand curve
Pij = The inflation-adjusted gas price in month i and year j
Yij = Real per capita disposable personal income in month i and year j
a, b = Coefficients on price and income
ei = A dummy variable for the month of the year to control for seasonal variation (there are actually eleven of these, one for each month January through November; they’re one if it’s the month in question and zero otherwise); this is called “seasonal fixed effects”
eij = A mean-zero random error term for month i and year j.

This way of specifying our model is called “linear”. This isn’t the only way to do it — at the end I’ll mention another way called “double log” that has some advantages. But for now, we’re ready to collect some data and run a regression.

Go Find the Data
The best source for data is always official government sources. Here’s the data we’ll use for this:

1. Gallons of gas demanded: We’ll use data from the U.S. Energy Information Administration. It’s called “product supplied”. The numbers are in barrels, so you’ll have to multiply them by 42 to convert them to gallons:

http://tonto.eia.doe.gov/dnav/pet/hist/mgfupus1m.htm

2. Gas prices: We’ll use numbers from the U.S. Bureau of Labor Statistics here. It’s from their “average price data” series, and it’s the monthly retail price of gas:

http://www.bls.gov/cpi/home.htm

3. Income: We’ll use data from the U.S. Bureau of Economic Analysis for this one. It’s called “disposable personal income”, and it comes from line 26 on Table 2.1 from the National Income and Product Account (NIPA) tables:

http://www.bea.gov/national/nipaweb/SelectTable.asp?Selected=N#S2

4. Something to inflation-adjust prices and income: For this one, we’ll use the implicit price deflator for GDP from the Bureau of Economic Analysis. It’s on line 1 of NIPA Table 1.1.9:

http://www.bea.gov/national/nipaweb/SelectTable.asp?Selected=N#S1

5. Population figures to turn gallons and income into per capita figures: This is a hard one, because we need monthly figures and the U.S. Census Bureau only produces annual figures. Also, it’s hard to piece together a consistent series from before and after each decennial census. Thankfully I’ve done the hard work for you — the Excel files below include a monthly population series I put together myself.

Once you’ve compiled these data in columns in an Excel sheet, you’re ready to run your regression. When you do, you should find something like this:

For 2000-2007, the coefficient on gas prices should be about -1.07, and the coefficient on income should be about 0.0007. Using the formula for price elasticity of E = (Average price over the period/Average quantity over the period)*(price coefficient), that implies a price elasticity of demand of about -0.048 and an income elasticity of about 0.51.

And that’s about what we’d expect. We know short-run demand for gas is inelastic, and has a negative relationship with prices. And we know that income should be positively related to gas demand, which it is.

The above method is based on a well-known 2006 study from Hughes, Knittel and Sperling which you can read at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=930730.

For those who’d like to see the final product, here are Chamberlain Economics’ own elasticity estimates for gasoline. The first file uses the linear specification above. The second one uses a “double log” specification, which basically takes the log of the data. The big advantage of the latter is that the regression coefficients are also the price and income elasticities, which is handy:

Price elasticity of demand for gasoline: Linear model.
Price elasticity of demand for gasoline: Double log model.

If you want the full data as STATA files instead, here they are:

Price elasticity of demand for gasoline: Linear model.
Price elasticity of demand for gasoline: Double log model.

Questions? Give us a call here.