- The BEA releasedinitial estimates of GDP for 2nd quarter of 2014 and revised estimates of GDP andGDI for the 1st quarter.
- Our graph is here.
- Jim Hamilton has lots of graphs. Probably the most relevant part of his post:
Our Econbrowser Recession Indicator Index, which uses today’s data release to form a picture of where the economy stood as of the end of 2014:Q1, jumped up to 24.1%, reflecting the sharp drop in GDP in the first quarter. Our policy of always waiting one quarter for data revisions and trend recognition before calculating the index is looking particularly wise given the roller coaster ride the BEA has put us through in their different versions of the 2014:Q1 estimates. And our index, unlike the BEA figures, is never revised. According to our gauge, the Q1 drop and slow first half growth were disturbing, but not the start of a recession– our indicator would have to rise to 67% before we would declare that a new recession had begun.
- The Upshot has graphs too. I’m not convinced that showing year-over-year changes in growth is that informative, but they have the pretties bar charts you’ll ever see.
- Also, the FOMC released its policy statement on interest rates. FT Alphaville has a summary and short analysis.
Our GDP graph (real GDP and real GDI) has been updated with today’s data release and is available at http://pseudotrue.com/gdp. GDP now runs through 2nd quarter 2014 and GDI through 1st quarter. (Initial GDI is released about a month after initial GDP.) GDP grew at an annual rate of 4% in the 2nd quarter, and fell at an annual rate of 2.1% in the first quarter (This number has been revised since the last report and is a smaller decrease than the last estimate.)
Remember that the estimate of 2.1% is only based on a partial dataset and it will be revised several times as more data are reported. (This revision process is routine and follows every GDP report.) When these new estimates are released, we will update our graph.
You can read the BEA’s summary of today’s release here: http://bea.gov/newsreleases/national/gdp/2014/gdp2q14_adv.htm
Our inflation graph (core and raw CPI) has been updated with today’s data release: http://pseudotrue.com/cpi.
I’m starting to work on some computationally demanding projects, (Monte Carlo simulations of bootstraps of out-of-sample forecast comparisons) so I thought I should look at Julia some more. Unfortunately, since Julia’s so young (it’s almost at version 0.3.0 as I write this) a lot of code still needs to be written. Like Random Number Generators (RNGs) that work in parallel. So this post describes an approach that parallelizes computation using a standard RNG; for convenience I’ve put the code (a single function) is in a grotesquely ambitiously named package on GitHub: ParallelRNGs.jl. (Also see this thread on the Julia Users mailing list.)
A few quick points about RNGs and simulations. Most econometrics papers have a section that examines the performance of a few estimators in a known environment (usually the estimators proposed by the paper and a few of the best preexisting estimators). We do this by simulating data on a computer, using that data to produce estimates, and then comparing those estimate to the parameters they’re estimating. Since we’ve generated the data ourselves, we actually know the true values of those parameters, so we can make a real comparison. Do that for 5000 simulated data sets and you can get a reasonably accurate view of how the statistics might perform in real life.
For many reasons, it’s useful to be able to reproduce the exact same simulations again in the future. (Two obvious reasons: it allows other researchers to be able to reproduce your results, and it can make debugging much faster when you discover errors.) So we almost always use pseudo Random Number Generators that use a deterministic algorithm to produce a stream of numbers that behaves in important ways like a stream of independent random values. You initialize these RNGs by setting a starting value (the “pseudo” aspect of the RNGs is implicit from now on) and anyone who has that starting value can reproduce the identical sequence of numbers that you generated. A popular RNG is the “Mersenne Twister,” and “popular” is probably an understatement: it’s the default RNG in R, Matlab, and Julia. And (from what I’ve read; this isn’t my field at all) it’s well regarded for producing a sequence of random numbers for statistical simulations.
But it’s not necessarily appropriate for producing several independent sequences of random numbers. Which is vitally important because I have an 8 core workstation that needs to run lots of simulations, and I’d like to execute 1/8th of the total simulations on each of its cores.
There’s a common misconception that you can get independent random sequences just by choosing different initial values for each sequence, but that’s not guaranteed to be true. There are algorithms for choosing different starting values that are guaranteed to produce independent streams for the Mersenne Twister (see this research by one of the MT’s inventors), but they aren’t implemented in Julia yet. (Or in R, as far as I can tell; they use a different RNG for parallel applications.) And it turns out that Mersenne Twister is the only RNG that’s included in Julia so far.
So, this would be a perfect opportunity for me to step up and implement some of these advanced algorithms for the Mersenne Twister. Or to implement some of the algorithms developed by L’Ecuyer and his coauthors, which are what R uses. And there’s already C code for both options.
But I haven’t done that yet. I’m
Instead, I’ve written an extremely small function that wraps Julia’s default RNG, calls it from the main process alone to generate random numbers, and then sends those random numbers to each of the other processes/cores where the rest of the simulation code runs. The function’s really simple:
function replicate(sim::Function, dgp::Function, n::Integer) function rvproducer() for i=1:n produce(dgp()) end end return(pmap(sim, Task(rvproducer))) end
That’s all. If you’re not used to Julia, you can ignore the “::Function” and the “::Integer” parts of the arguments. Those just identify the datatype of the argument and you can read it as “dgp_function” if you want (and explicitly providing the types like this is optional anyway). So, you give “replicate” two functions: “dgp” generates the random numbers and “sim” does the remaining calculations; “n” is the number of simulations to do. All of the work is done in “pmap” which parcels out the random numbers and sends them to different processors. (There’s a simplified version of the source code for pmap at that link.)
And that’s it. Each time a processor finishes one iteration, pmap calls dgp() again to generate more random numbers and passes them along. It automatically waits for dgp() to finish, so there are no race conditions and it produces the exact same sequence of random numbers every time. The code is shockingly concise. (It shocked me! I wrote it up assuming it would fail so I could understand pmap better and I was pretty surprised when it worked.)
A quick example might help clear up it’s usage. We’ll write a DGP for the bootstrap:
const n = 200 #% Number of observations for each simulation const nboot = 299 #% Number of bootstrap replications addprocs(7) #% Start the other 7 cores dgp() = (randn(n), rand(1:n, (n, nboot)))
The data are iid Normal, (the “randn(n)” component) and it’s an iid nonparametric bootstrap (the “rand(1:n, (n, nboot))”, which draws independent values from 1 to n and fills them into an n by nboot matrix). Oh, and there’s a good reason for those weird “#%” comments; “#” is Julia’s comment character, but WordPress doesn’t support syntax highlighting for Julia, so we’re pretending this is Matlab code. And “%” is Matlab’s comment character, which turns the comment green.
We’ll use a proxy for some complicated processing step:
@everywhere function sim(x) nboot = size(x, 2) bootvals = Array(Float64, nboot) for i=1:nboot bootvals[i] = mean(x[x[:,i]]) end confint = quantile(bootvals, [0.05, 0.95]) sleep(3) #% not usually recommended! return(confint < 0 < confint) end
So “sim” calculates the mean of each bootstrap sample and calculates the 5th and 95th percentile of those simulated means, giving a two-sided 90% confidence interval for the true mean. Then it checks whether the interval contains the true mean (0). And it also wastes 3 seconds sleeping, which is a proxy for more complicated calculations but usually shouldn’t be in your code. The initial “@everywhere” is a Julia macro that loads this function into each of the separate processes so that it’s available for parallelization. (This is probably as good a place as any to link to Julia’s “Parallel Computing” documentation.)
Running a short Monte Carlo is simple:
julia> srand(84537423); #% Initialize the default RNG!!! julia> @time mc1 = mean(replicate(sim, dgp, 500)) elapsed time: 217.705639 seconds (508892580 bytes allocated, 0.13% gc time) 0.896 #% = 448/500
So, about 3.6 minutes and the confidence intervals have coverage almost exactly 90%.
It’s also useful to compare the execution time to a purely sequential approach. We can do that by using a simple for loop:
function dosequential(nsims) boots = Array(Float64, nsims) for i=1:nsims boots[i] = sim(dgp()) end return boots end
And, to time it:
julia> dosequential(1); #% Force compilation before timing julia> srand(84537423); #% Reinitialize the default RNG!!! julia> @time mc2 = mean(dosequential(500)) elapsed time: 1502.038961 seconds (877739616 bytes allocated, 0.03% gc time) 0.896 #% = 448/500
This takes a lot longer: over 25 minutes, 7 times longer than the parallel approach (exactly what we’d hope for, since the parallel approach runs the simulations on 7 cores). And it gives exactly the same results since we started the RNG at the same initial value.
So this approach to parallelization is great… sometimes.
This approach should work pretty well when there aren’t that many random numbers being passed to each processor, and when there aren’t that many simulations being run; i.e. when “sim” is an inherently complex calculation. Otherwise, the overhead of passing the random numbers to each process can start to matter a lot. In extreme cases, “dosequential” can be faster than “replicate” because the overhead of managing the simulations and passing around random variables dominates the other calculations. In those applications, a real parallel RNG becomes a lot more important.
If you want to play with this code yourself, I made a small package for the replicate function: ParallelRNGs.jl on GitHub. The name is misleadingly ambitious (ambitiously misleading?), but if I do add real parallel RNGs to Julia, I’ll put them there too. The code is still buggy, so use it at your own risk and let me know if you run into problems. (Filing an issue on GitHub is the best way to report bugs.)
P.S. I should mention again that Julia is an absolute joy of a language. Package development isn’t quite as nice as in Clojure, where it’s straightforward to load and unload variables from the package namespace (again, there’s lots of code that still needs to be written). But the actual language is just spectacular and I’d probably want to use it for simulations even if it were slow. Seriously: seven lines of new code to get an acceptable parallel RNG.
A grad student emailed me about problems she’s running into with MLE in R. In short, the parameter estimates are fairly sensitive to the optimization’s starting values, which is a problem. Since it might be useful for other people, this is the reply I sent (any other suggestions would be welcome!):
Those problems can show up when the likelihood function is relatively flat or when the likelihood has local maxima. You can diagnose these problems a few ways.
- Plot the likelihood function. If it only depends on a single parameter, this is really easy. If it depends on several parameters, you’ll want to fix all but one or two of the parameters and plot it as a function of the remaining one or two. (And then repeat this process for different values of the other parameters, and repeat it while fixing and varying other parameters.)
- Check it for simulated data with *lots* of observations (e.g. millions). When there are a lot of observations, the likelihood should be quadratic around the true parameter values, so there should be fewer problems with local maxima.
- Do them both: make plots with lots of observations and verify that the likelihood is approximately quadratic around the true parameter value.
If you’re supplying a gradient function it can cause problems as well. You can compare the analytical gradient to the output from
numericDeriv to verify that your gradient is correct. (
numericDeriv is for R code, which is what the student asked about specifically; other languages should have similar functions.)
The unemployment graphs have been updated with June, 2014’s data. Sorry that these are a little late!