Identities, Parameters and Regressions

A reader comments:

Your final jab in this post regresses the state output gap on the fiscal gap. You then conclude that there is a positive relation between the two and that this somehow implies that a reduction in gov’t spending is a drag on the economy. I’ll just point out that …. that gov’t spending is a component of GSP. Of course they’re positively related.

I think this comment reflects a commonplace confusion between identities, functional relationships, and reduced form coefficients.

For instance, consider that at the industry level for a state, such as in data reported by BEA here, total private (agriculture, mining, through manufacturing through other services) and government has to sum to total gross state product (in current lexicon state GDP). So, one would think that regressing total output on government output has to yield a positive coefficient. Run the regression over the 2005Q1-15Q3 period (the entire available sample) for Wisconsin:

GDPWI,t = 262893.4 – 0.215GOVWI,t + ut

Adj-R2 = -0.02. Bold denotes significance at 10% msl using HAC robust standard errors.

Lest you think this proves higher government spending causes less output, well, consider the same regression for Kansas.

GDPKS,t = 18829.71 + 5.895GOVKS,t + ut

Adj-R2 = 0.34. Bold denotes significance at 10% msl using HAC robust standard errors.

This is a lesson I learned as an undergraduate — do not appeal to an accounting identity for information about a coefficient.

Let’s turn to an example that is more familiar. We all know the expenditure side definition of GDP from macro:

Y ≡ C + I + G + EX – IM

I stress in my undergraduate courses that one cannot use identities to explain how the world works, or more concretely, when G changes, how does Y. For that, one needs a model.

So, the hapless regression-runner might regress Y on G, asserting one must get a positive coefficient since G is by construction a component of Y. But in fact different theories yield different implied coefficients.

Suppose aggregate supply is given by Y = Yn = ΦF(K,N), and aggregate demand is given by Y = fn(G, T, M/P), where K, N are given. Then the correlation between Y and G is … zero!

Suppose P is predetermined within a period, and aggregate demand takes the same form as above. Then the correlation between Y and G is positive, but not necessarily one (it’ll depend on the marginal propensity to consume, interest sensitivity of investment, interest sensitivity of money demand, etc.).

Suppose P depends positively on the output gap (i.e., a Phillips Curve holds), such that P = Pe + θ(Y-Yn). Then the correlation is positive, but (holding all else constant) smaller than that in the previous example.

Suppose P depends positively on the output gap, and the fiscal authorities rely on a “fiscal rule” that achieves counter-cyclical stabilization, e.g., G = φ(Y-Yn), φ < 0. Then the implied correlation between Y and G is now negative.

One could in principle estimate the reduced form coefficient in the first three cases (after accounting for omitted variables, etc.); the parameters (e.g., θ) would require determining an appropriate instrumental variable. The correlation can always be calculated; whether it’s meaningful is another question.

Bottom Line: Identities do not tell you about behavior. Inferring causality is hard, but if one wants to tell a story, one has to try to deal with the data in an intelligent way.

25 thoughts on “Identities, Parameters and Regressions

  1. 2slugbaits

    Menzie I can see where some folks might misunderstand and not catch the fact that your regressions are using the same “t” on both sides of the regressions, so these are contemporaneous “relationships”…or rather “non-relationships.”

  2. PeakTrader

    It’s supported by the economics literature:

    “Several studies find that reductions in government spending “can have expansionary effects, since they can contribute to a consumption and investment boom owing to altered expectations regarding future taxation.”

    A 2002 study of U.S. data find that the effect of debt-financed spending increases was somewhat positive, but the multiplier effect was much less than one.

    A 2004 IMF study of recessions in advanced economies found that “multipliers are unlikely to exceed unity.”

    A 2006 study of U.S. data found the effect of “fiscal expansion appears insignificant on aggregate demand and economic activity.”

    In December 2008, the National Bureau of Economic Research: “The best fiscal policy to stimulate the economy,” they report, “is a deficit-financed tax cut,” and “the long term costs of fiscal expansion through government spending are probably greater than the short term gains.” … “government spending shocks crowd out both residential and non-residential investment,” while “the [positive] response of consumption is small and only significantly different from zero on impact””

    My comment: An increase in government spending may be most effective to raise output when the national debt is low and negative when the national debt is too high.

    We’ve had accommodative monetary policy (an explosion in the Fed balance sheet) and expansionary fiscal policy (an explosion in the national debt). Yet, we had very slow growth.

  3. MSJ

    You seem to be trying to use the hammer of regression models … to open a wine bottle. This makes zero sense

    (The regression just lines up coincident timing … but this makes zero sense in behavioral, long history, massively complex systems, right? The mind and logic is 100x the power.)

    The right global econ perspective is likely something like:

    (1) — When output is suboptimal, unlimited fiscal spending is free (or even better than free). See WWII. What did we spend and owe? Yet we came out rich. Why? Hmm… Spending makes us rich, right?

    (2) — When output is nearing limits, fiscal spending “crowds out more efficient private sector ideas” to some degree. Since horrible economists came to rule govt with all the wrong ideas ;in the 1970s and then 1980s (lower cap gains taxes bc economies need capital? Seriously? Bc capital is mobile? Are you guys idiotic by design?) We have likely left massive infrastructure and govt research low hanging fruit. Private capital does “social networking websites” very well … but not nuclear energy, or the grid, or new modes of transport, or schools, or roads.

    (3) — Maybe today we have lots of good runway with smart fiscal spending. What do we learn from Nazi Germany and 1940s Japan? … That govt $ spent efficiently yields near unlimited productivity! Now US culture is not as efficient as those “order above all” mental cultures. But see how important this is – those sick Germans almost owned the world with better economics! We have forgotten, and people lives are ruined by the selfish subconscious desire for scarcity (and the ranking of “rich”). We are so lucky to have had Keynes and Roosevelt and Churchill and Eisenhower on THAT JOB. Today, you fools would lose it bc “too much debt, cannot afford to win” right?

    How petty and wrong is that thinking? So then, some infrastructure areas here can always take spending and be simply unfathomably good investments for Thomas Jefferson’s startup USA Inc. Larry Summers is good here on understanding that – lack of investment in infrastructure is a much more expensive form of debt. Or Freddie Mercury and David Bowie give you the whole truth –

    1. Peaktrader

      MSJ, so, your plan is to spend, tax, and destroy. Why bother with rigorous economic models to understand how economies really work.

  4. Anonymous

    Ok, but come on, is that really what the commenter meant? If in time t a government goes in debt and does stimulus spending at 20 percent of GDP, couldn’t one realistically assume that in time t GDP would be greater than it would if the stimulus had not occurred, given that G is in the formula?

    If you can’t even assume that than should anyone really be arguing for stimulus policies?

    1. Menzie Chinn Post author

      Anonymous: Sure, but when does government spending to GDP rise by 20 percentage points? In 2008Q4, government consumption and investment was 21.0%; by 2009Q3 (as the ARRA kicked in), it peaked at 21.6%, and then declined. And it’s been declining ever since. Hence, your example is completely irrelevant.

  5. Julian Silk

    Dear Menzie,

    I agree with all this, but think you would have a more conclusive result, if one with poorer R-squares, if you regressed changes in GSP vs. changes in state government spending. After all, it was the cuts in Kansas government spending that were supposed to usher in this glorious era. It hasn’t quite worked out, has it?


  6. Rick Stryker


    Your conclusion that you can’t infer parameters from identities is certainly correct . But your regression analysis does not establish that point.

    As you put it: “So, one would think that regressing total output on government output has to yield a positive coefficient.” If you did think that, you’d be right. The negative coefficient you got for Wisconsin is an artifact of the small sample size you have. You are regressing the levels of two non-stationary variables on each other, variables that would have a positive trend with sufficiently long data. If you had a long enough data set, you would always get a highly statistically significant positive coefficient in that Wisconsin regression.

    By the way, there is a typo in the results: the intercept of the Wisconsin and Kansas regressions are switched around.

    1. Menzie Chinn Post author

      Rick Stryker: Yes, in a big enough sample … but for those of us who’ve worked in cointegration and the “great ratios” (e.g., C/Y), it’s amazing how big the sample has to be.

      Thanks for pointing out the typo.

      1. Rick Stryker


        I’m curious why you think the Wisconsin series are cointegrated though (if that’s what you meant). Neither the Phillips-Ouliaris Pz nor Pu tests indicate cointegration, although that may be a small sample problem. If they aren’t cointegrated, then the regression you ran is spurious. Even so, you’d still expect to get a positive, highly significant coefficient in large enough samples.

        1. Menzie Chinn Post author

          Rick Stryker: I think you are kind of missing the point I was aiming at Professor Russell. His point was if Y ≡ Z + W, then in a regression it must be the case that there is a positive correlation (irrespective of correlation, etc.) between Y and Z. I used GDP and GOV in a linear form because that was what was consistent with an identity. If I had my ‘druthers, I would have estimated in log first differences given what I know (by the way, Johansen does reject the no cointegration null at 10% for levels if allowing for quadratic deterministic trend). But that would not have driven home the incorrectness of Professor Russell’s assertion.

          1. Rick Stryker


            No, I didn’t miss the point. I began my initial comment in this thread with “Your conclusion that you can’t infer parameters from identities is certainly correct . But your regression analysis does not establish that point. ” My point is that the regression you selected to drive home your point doesn’t actually make your point, since you selected a regression that would get a positive coefficient if you just had enough data.

            By the way, if you really believed that the levels of the data had a quadratic deterministic trend, then that would imply in general a linear trend in the cointegrating relation in the Johansen framework. However, you did not include a linear trend in your regression. I’d also point out that if you really do think this is a cointegrating regression, you really can’t just calculate HAC standard errors as you did, since the t stats are generally invalid in this case. Also, I’m not sure why to report R2, since it goes to 100% as the sample size gets large. The t stat and R2 points are true whether the underlying regression is spurious or cointegrating.

          2. Menzie Chinn Post author

            Rick Stryker: I report the same stats all the time. I understand the point. In the futre I’ll make sure to put in a caveat about inference with integrated series (I’d thought people were bored by this topic, but your comment tells me that’s not the case).

          3. Menzie Chinn Post author

            Rick Stryker: By the way, what’s your thoughts on the revelation of Donald Trump as the face of the Republican Party? I must confess that I am unsurprised.

    2. Julian Silk

      Dear Rick,

      From what I can tell, the sample would have to be enough so that the other variables, such as investment in particular (or investment over GDP or some version of the logs of the two), would have to revert to roughly mean values or higher than mean values on occasion in this case. You’d need this so the variables you were less interested in would be roughly stationary, so that nonstationary government would show up positively vs. nonstationary GDP. In my energy work that was published with Fred Joutz in Energy Economics in 1997, 45 years was nowhere near enough to allow variables you might hope to be stationary to become so, if you allow for structural shifts in the equation. We modeled residential electricity consumption vs. income, from 1949 to 1993, and the variable that you might hope is stationary is natural gas prices. We used dummy variables, because you can’t model everything, and it makes a big difference.


      1. Rick Stryker

        Thanks Julian. I’ll look at your paper.

        I was more focused on the technical aspects of the Wisconsin regression. If the Wisconsin variables were cointegrated, then yes, the Monte Carlo evidence suggests it can take a lot of data to converge to the true parameters, which would represent the long-run relationship between the variables.

        Here I think the situation may be a bit different. Although I can see that the variables might be cointegrated, I would not be surprised if they weren’t, especially as this is a bit of an odd regression, with levels rather than log of levels of trending non-stationary variables. The variables don’t seem to be cointegrating according to standard tests. In this case, the parameter that will be estimated by the regression does not represent any economic relationship, in the short run or the long run, but is rather spurious. But even if it is spurious, the estimated parameter will be positive and highly statistically significant given enough data.

        1. Julian Silk

          Dear Rick,

          My hope is to get started on something that uses the cointegration techniques on another issue shortly. But here it’s an interesting question what test one uses to look at this. With Johansen-Juselius tests, there is always a power problem, which people have agreed to ignore, for variables that are near-integrated. A nice look at this may be found in David Hendry and others make the argument that you want to start with this very large data set of possible variables and narrow it down, and so they like the Johansen-Juselius test. But my impression is that the conclusion that you want to have supporting evidence besides just the test is quite acceptable to the OxMetrics people, and maybe to Johansen and Juselius themselves. (I found her very easy to get along with, while he struck me as a forbidding Norseman.) Since the log is a monotonic transformation, if the levels are really cointegrated, with any degree of strength, it would seem the logs are too, but again, it hasn’t really been looked at.


          1. Rick Stryker

            Hi Julian,

            I believe the paper you are referring to is concerned with the size of the Johansen test rather than the power. The paper finds that the Johansen test tends to find cointegration too often, which is consistent with the monte carlo evidence reported in Cheung and Lai (1993) in “Finite Sample Sizes of Johansen’s Likelihood Ratio Tests for Cointegration” in the Oxford Bulletin. That’s another reason why I tend to believe the series that Menzie examined are not cointegrated–the Johansen test is biased to find it if it is not there in finite samples. That fact plus the weak statistical evidence (10% confidence level) and the requirement to have a quadratic trend in the data suggest that there is no cointegration. I don’t think Menzie is necessarily arguing for cointegration but, if it’s not there, he has estimated a spurious regression in which all parameter estimates and test statistics are meaningless. I think your earlier suggestion to have run a regression in differences was a good one.

            Power and size considerations are of course a big problem with all these unit root and cointegration tests. The problem with Johansen beyond that standard difficulty is that it’s just very easy to make a specification error. I’d agree that Johansen and Juselius would want to bring in other evidence besides the test results. That’s what they did in their 1992 paper that investigated PPP and UCIP. The max eigenvalue and trace tests disagreed on the number of cointegration vectors and J&J argued from considerations outside the tests to determine which test to believe.

            Not sure if you are famiiiar already, but when considering what cointegration tests to use in terms of finite sample bias, I’d recommend Haug’s 1996 Journal of Econometrics article “Tests for Cointegration: A Monte Carlo Comparison,” which investigates a range of cointegration tests in terms of both size and power.

            Hope your research goes well.

            Note: this comment was rejected by the blog software as already having been made, so I apologize if it’s repeated

  7. Levi Russell

    Of course, I clarified this comment in the previous post. You are behaving quite disingenuously, Menzie.

    1. baffling

      levi, a word of advice. an untenured faculty member should really limit creating controversy in the blogosphere, and instead focus on producing quality research and peer reviewed publications. your time will come when you can take over the blogosphere, but now is not the time. you should have mentors at your institution to help guide you in this endeavor. for an untenured faculty member, the blogosphere is like the rest of social media for your students- a disaster waiting to happen. again a friendly piece of advice. content on the internet never disappears…and in due time you will be evaluated for tenure. let your content be peer reviewed papers, not internet banter.

  8. XO

    Please listen to Baffling, that advice is so so true. Life before and after tenure is very different. Everything before tenure should be focused on peer reviewed publications. Publish early and often. After tenure, you can build a broader spectrum of outlets for your ideas. Anything not peer reviewed now is just taking away from what you need to be focusing on in this 5/6 year period. And, there is no right to be forgotten in the USA….

  9. Rick Stryker


    To answer your question, Trump may represent the face of the Republican party, at least for the time being, but he doesn’t represent its once and (I hope) future conservative ideals.

    Trump is an independent candidate who successfully mounted a third party run from within the Republican party. Always the opportunist, Trump realized he couldn’t succeed as in independent candidate and needed to be nominated by one of the parties. I think he could have gone either way on the choice he made, adopting his persona as needed. Trump said in a Playboy interview in 1990 that he’s a much more natural fit for the Democratic party if he ran for President. But Trump must have realized that he could not have gotten the nomination as a Democrat this year. Hilary Clinton was too much the presumptive candidate. Besides, the Democratic rules award delegates in proportion to the votes received and the super delegates would have sided with Hilary. Under the Democratic primary rules, Trump could not have parlayed 37% of the vote he received initially into a majority of the delegates, thereby making him the frontrunner and causing more people to jump on the bandwagon.

    The Republican rules are much more amenable to nominating a candidate not favored by the party, given that the winner-take-all or winner-take-most states magnify delegate support, and given that there are no superdelegates to express countervailing party views. Plus, the field was wide open on the Republican side. Certainly many Republicans saw the very real possibility that Trump could pull it off, but I must say that I am shocked that he managed to do it and amazed that so many Republicans voted for him, given that he appears to be a catastrophe for the Republican party.

    But I think he’s potentially a serious problem for the Democratic party too. Trump is not planning to run as a traditional Republican, someone who starts with his conservative base and then enlarges his coalition by bringing in independents and conservative Democrats, in order to win key purple states such as Florida, Virginia, and Ohio. In my view, Trump, no conservative, is planning to assemble a much different coalition. As Trump moves into the general election, he’ll begin to be more open about positions he’s been hiding in plain sight. Prominent anti-capitalist intellectual Slavoj Zizek noted that Trump is actually a centrist liberal, which, coming from a European leftist, means that Trump’s views are squarely in the center to left wing of the Democratic party on many issues. You may think Zizek is exaggerating, but consider: Trump said Bush lied and people died during a Republican debate. Trump has said that single payer health care works. Trump has endorsed the Obamacare mandate. Trump says that it’s the government’s responsibility to make sure everyone has health care. Trump’s foreign policy ideas are non-interventionist and isolationist. Just today, Trump signaled that he’s open to supporting a higher minimum wage, reversing his previous stance. Trump supports planned parenthood and many conservatives believe he’ll reveal his true beliefs about abortion in due time.

    These are hardly the positions of a traditional Republican, but, as Trump said today of his reconsideration of the minimum wage, I’m very different from most Republicans. Trump won’t run as a traditional Republican because he isn’t one. Instead, Trump will start with a base of Republicans who support his anti-immigration and protectionist policies. He’ll also start with a base from the liberal and moderate wing of the Republican party. Conservatives are free to join the coalition if they want, but on his terms–he’ll not do anything for them other than make a few perfunctory gestures. Trump will move left in general but not quite as far left as Hilary, attacking her from both the left and the right. From the left, Trump is already starting to make a play for the Sanders voters, saying that the Democratic primary rules are rigged against Sanders and his voters. Trump can appeal to the Sander voters by contrasting his alleged opposition to the Iraq war with Hilary’s vote for it. He can also emphasize his much less interventionist foreign policy views, his willingness to come to terms with Putin, his support for “fair trade,” and his support for many big government policies. Sanders voters also want an outsider and Trump will present himself as an outsider while labeling Hilary as the epitome of the corrupt insider. Trump probably can’t get a majority of Sanders voters, but he may get a lot of them. On the right, he’ll attack on gun control, immigration policy, trade, and, oddly enough, foreign policy. Trump will also challenge Hilary in the center and, importantly, challenge her blue collar Democratic support in key states.

    Those key states are New York, Pennsylvania, Ohio, Michigan, and perhaps a few others, the states with manufacturing industries that have been most hurt by international trade and globalization. If Trump can assemble some of the Sanders vote, the blue collar union vote, and the other components of his new coalition, he may be able to win those states. In this strategy, Trump does not necessarily plan on winning states like Florida or Virginia, although he will try to do so. Rather, if Trump can start with the states that Romney won, minus a few that he’ll lose because of opposition from Hispanics, but then add the big rust belt states, he’ll be able to get close to 270. He’ll then just need to add a win or two somewhere else to put him over the top.

    It’s not clear at this point whether Trump can do this. I think a landslide for Hilary is more likely at this point. But no one should underestimate Trump. If Trump does manage to win with this non-traditional strategy, he’ll tear the Republican party as we know it apart. But he’ll also tear the Democratic party as we know it apart. For Democrats who want to keep their party together, the choice will be clear: vote for Hilary. For conservatives, it’s not so clear. Do you vote for Trump or Hilary as the lesser of two evils? Do you stay home or do you write in a candidate? Or do you go all in and actively work for Hilary? This is the complex dilemma that many conservatives are wrestling with.

    I ask myself these questions a few times a day and I’m not sure what the right answer is. I wonder what other econbrowser readers plan to do about the Trump Question.

  10. baffling

    not sure if hilary is a candidate ,but Hillary will certainly be a formidable opponent. smart money would move forward with the garland nomination.

    Rick, i find an interesting observation, mostly of older conservatives, who argue that trump is not a true republican nor conservative. perhaps you are correct on conservative. but perhaps you are missing the bigger picture of the republican party. it seems to appear that the conservatives are simply becoming a cohort without a legitimate party affiliation. the republican party has changed, and not for the better. trump has revealed consistent, conservative viewpoints are not being embraced by the larger republican audience. they are angry, because they have been left behind. vocally, they blame democrats for their failures. but from an electorate perspective, many of these folks appear to be saying these traditional conservative leaders have hurt me, and i am rejecting those who continue to promote such policies. those who argue trump is tearing the republican party apart may be misunderstanding what the people are saying, which is simply an unhappiness with conservative leadership of the party. success of a traditional conservative candidate was completely nonexistent in this primary. without a single unifying conservative agenda, a splintered republican party may be much harder to hold together than the coalition of democratic groups on the other side.

    1. Rick Stryker


      I agree with what you are saying. More precisely, Trump’s brand of Republicanism is inconsistent with the party leadership’s version. But ~40% of Republican primary voters rejected the party’s version in favor of Trump’s version. Moreover, the other 60% were not able to consolidate around an alternative. There is a disconnect here.

      Trump seems to be already executing the strategy I suggested and getting some results. The Qunnipiac poll today showed that Trump and Hilary are neck and neck in Ohio, Florida, and, amazingly, Pennsylvania. There are a lot of undecided voters though. If the strategy keeps working, Trump will keep going. But if it doesn’t work, I have no doubt that Trump will find his inner conservative and switch to something more traditional. Trump needs to raise a lot of money–1.5 billion dollars from Republican donors. I’d like to see how he plans to do that if he keeps going with the third party triangulation strategy. Of course, nothing succeeds like success so if it looks like Trump is winning a lot of donors might overlook the ideology problems, especially if Trump assures them privately that he doesn’t really mean it.

      Hilary is already protecting her left flank by moving to support a public option in health care today.

      1. baffling

        “More precisely, Trump’s brand of Republicanism is inconsistent with the party leadership’s version.”
        therein lies the problem. who actually has ownership of the republican and conservative brands. in the past, one did not discriminate too much with this difference. but trump brought into the argument brand recognition, and wants ownership of some sort. if 2 bases of power form, trouble on that side. at least the convention should be interesting this year.

        and her name is Hillary not Hilary.

Comments are closed.