A real-time bubble estimate could be worth money, but this paper stops before proving the trading case. Krabbe builds the machinery, applies it to the Nasdaq real spread through December 2003, and separates that spread into a fundamental component and a bubble component. I read the paper twice because the central claim matters. No return series, no threshold, no position, no cost. The result is a state estimate.

A disclosure comes first. The NASDAQ Composite is not holdable, so we applied the same decomposition to QQQ and traded QQQ. Our exercise does not replicate the paper. Every dot-com estimate below remains specific to the Composite, using Krabbe's sample and Krabbe's construction of the real spread.

The model

A state space model combines an observed series with a hidden state. Krabbe's causal non-causal version contains two hidden states moving in opposite time directions. One depends on its own past, while the other depends on its own future. Their sum produces the observed series.

The fundamental component is causal, a Gaussian AR(1) with |rho_F| < 1. The bubble is non-causal, an alpha-stable AR(1): X^B_t = rho_B X^B_{t+1} + eps^B_t. Today's value therefore depends on tomorrow's, and the innovations are heavy-tailed. Krabbe's Figure 1 isolates the bubble process at rho = 0.9, alpha = 1, sigma = 0.5, beta = 0. Its shape is familiar. A large innovation arrives at some date tau. Viewed backward from tau, the process decays geometrically. In observed forward time, the same path becomes an exponential rise followed by a crash at tau. Lead dependence gives a statistical form to a path that appears to anticipate its own collapse.

Gouriéroux and Zakoïan introduced the convolution setup. Krabbe adds the inference machinery. Propositions 2 and 3 derive filtering and smoothing distributions for the non-causal case through what he calls a backward kernel and a forward function. Proposition 6 supplies new distributional results, including the closed-form characteristic function of Y_{t-p:t}.

The application uses CRSP monthly returns on the value-weighted portfolio of all Nasdaq stocks from January 1973 to December 2003, deflated by CPI. Krabbe models the real spread P_t minus D_t/R with R = 0.007 monthly. Table 3 gives the parameter set. The filtered bubble turns positive only near the sample's end, while the benchmark bubble remains positive throughout.

Two hidden states in one series

Krabbe shows that the convolution model can satisfy the standard rational expectations pricing condition E[B_{t+1}] = (1+R) B_t. The restrictions are alpha < 1 and beta = 1, making the bubble component non-negative, together with the exact link rho_B^{alpha-1} = 1 + R. Once the discount rate is fixed, the tail index and bubble persistence are tied together.

The restriction has bite, and the result is a good one. An unusual implication comes with it. Proposition 4 states E[X^B_t | X^B_{t-1}] = sign(rho_B)|rho_B|^{alpha-1} X^B_{t-1}. Krabbe observes that for alpha < 2, this conditional expectation differs from the best linear predictor under the scale criterion.

Proposition 5 identifies that predictor when rho_B not equal to 0 with beta = 0, or when rho_B in (0,1) with alpha not equal to 1. For 0 < alpha <= 1, the optimal linear coefficient is exactly 0 if |rho_B|^alpha lies below 1/2. Above 1/2, it jumps to exactly 1/rho_B. At exactly 1/2, either value is possible and the coefficient is indeterminate. At rho_B = plus or minus 1/2 with alpha = 1, an entire interval qualifies. A bang-bang predictor. Anyone trading the linear version would find the signal in that discontinuity.

Krabbe defines the model with transition kernels. When the non-causal kernels have densities, he recasts it as an ordinary causal state space model and uses standard filtering. Without those densities, he directly derives filtering and smoothing for the non-causal case. Filtered states use information available through t. Smoothed states use the whole sample. For a trader, the distinction carries the argument.

Estimation uses the Knight and Yu (2002) empirical characteristic function estimator, with block size p = 1 and weight w(u) = exp(-u'u). The characteristic function of Y_{t-p:t} is available in closed form, although the density generally is not. One especially neat case appears at alpha = 1 and beta = 0: Gaussian plus Cauchy is exactly Voigt, with a closed-form density obtained through the Faddeeva function.

What Krabbe finds on Nasdaq

The modelled series is the real spread P_t minus D_t/R. Krabbe sets R = 0.007 monthly, taking mean monthly nominal return of 1.1% and subtracting mean monthly inflation of 0.4%.

For the convolution model, rho_B is tied to alpha and beta = 1. The estimates are mu_F = -0.148, rho_F = 0.936, sigma_F^2 = 0.278, rho_B = 0.877, alpha = 0.947, sigma_B = 0.003. With alpha below one, the bubble component lacks an unconditional mean. Conditional forecasts still exist, the result established in Proposition 4.

The benchmark uses the same non-causal stable AR with a constant considered by Gouriéroux and Zakoïan. It imposes rho_F = 0 and sigma_F^2 = 0. Krabbe reports mu_F = -4.701, rho_B = 0.962, alpha = 0.821, sigma_B = 0.012. He also notes that the benchmark bubble accumulates even more slowly.

Its constant lies near the sample minimum spread of -4.368. With no fundamental process, the implied bubble becomes X^B_t = Y_t minus mu_F and must remain non-negative, forcing the constant toward that minimum. The benchmark consequently labels the entire 1973 to 2003 sample a bubble. Under the convolution model, the filtered bubble becomes positive only at the sample's end. By then, the two estimates are close to identical in size.

The benchmark's positive-throughout bubble conflicts with the evidence in Phillips et al. (2011). Krabbe gives that conflict as his reason for judging the convolution model more realistic. The preference rests on consistency with outside evidence rather than a formal test between the models.

A practitioner should spend time on the filtered and smoothed paths. Both identify the episode. At the sample's end, the real-time filtered bubble exceeds the smoothed estimate. Looking back, the model assigns the late rise in the spread to both the fundamental and bubble components. The retrospective signal is milder than the reading available at the time, and nobody measures the value of that asymmetry.

Full-sample parameters drive the real-time result

Table 3 estimates the parameters once on the full December 1973 to December 2003 sample, including the episode the filter is meant to detect. Section 4.2 formulates estimation on a single sample running from 1 to T. Section 6 then runs the filter in Figure 7 with those same estimates. The paper describes no recursive re-estimation. Its state recursion sees only observations available through each date, while its parameters come from the completed sample. The claim of real-time bubble detection therefore applies to a filtered state path calculated with full-sample parameters.

Identification imposes another limit, and Krabbe's simulations provide the clearest evidence. The model tries to infer two unobserved components from one observed series. Krabbe writes that the estimated states are "reasonably close to the true states overall; there are periods in which they are close to the true ones, and there are periods in which they are far away." He then adds a second noisy observation of the fundamental and repeats the experiment. The extra series keeps the estimated states close to the true states at all times. At T = 500, rho_F's simulation standard deviation drops from 0.015 to 0.007, while mu_F's falls from 0.687 to 0.344.

Identification binds before sample size does.

No equivalent auxiliary series exists for real stock prices. The Nasdaq application therefore uses one observed series, the version whose simulated state estimates track the truth only intermittently. Every convolution-model figure quoted here comes from that version.

The simulation study is clean. Krabbe generates 10,000 simulated samples at T = 125, 250 and 500, re-estimating the model each time to see whether it recovers the true parameters. Standard deviations shrink as T grows. Yet the exercise fixes alpha = 1 and beta = 0, unlike the Nasdaq parameterisation of alpha near 0.947 and beta = 1. Krabbe reports no standard errors with the Nasdaq estimates.

He states plainly that the asymptotic properties of the empirical characteristic function estimator for this model remain future research. Consistency and asymptotic normality are assumed rather than established here. The observation kernel also has no density because Y_t is a deterministic sum, so inference substitutes an approximating Gaussian observation kernel. The approximation error is not quantified. Filtered and smoothed expectations are truncated at 10^6 because moments may not exist for alpha <= 1, and I did not find a sensitivity check for that bound.

From decomposition to trade

The paper never places a trade.

Its empirical analysis ends with the decomposition. Three elements would be needed before it became a rule:

An econometrics paper can reasonably stop at the decomposition. The abstract promises to "illustrate the usefulness" of these models. In this paper, usefulness means separating the spread into components rather than producing a trading rule.

One episode, one asset, one filter run with full-sample parameters. The pricing-consistency restriction rho_B^{alpha-1} = 1 + R is a genuine result. Propositions 2, 3 and 6 also provide the filtering, smoothing and closed-form characteristic function missing from the earlier literature. A recursive version would change my view of the trading case: re-estimate the parameters using data through each date only, then show that the filtered bubble still appears before the crash. Until someone runs that exercise, the real-time claim belongs to the filter.