Theoretical Frontier Extensions And Applications Of Moderate Deviations From A Unit Root

Moderate deviations from a unit root,proposed by the famous Econometrician Peter Phillips,have attracted much recent attention.The so-called moderate deviation from a unit root is actually an autoregressive model where the autoregressive root is greater than unity but its deviation from unity decreases as the sample size increases.Such triangular array data processes have been shown to capture the explosive bubble behavior in many economic and financial time series.Moreover,it can be useful in conveying the severity of bubbles to policy makers.Therefore,how to detect the moderate deviation has become a significantly important issue,both theoretically and practically.This thesis concentrates on developing new theory and methods for the moderate deviation from a unit root,and applying these proposed methods to some substantive economic issues.The moderate deviation from a unit root is closely related to the unit root process.Based on the relationship between the autoregressive root and unity,we establish a unified framework which consists of the standard unit root process,the local-to-unity process,the local-to-moderate deviation process,the moderate deviation process,and the explosive process.This unified framework takes full advantage of the characteristics of the moderate deviation process,and links the above autoregressive processes from the aspects of model setting,data characteristics,convergence speed and limiting distribution.To our best knowledge,this is the first paper in the literature to provide such a general framework.The results show that the moderate deviation process effectively smoothes the discontinuity of the asymptotic properties of coefficient estimation errors under different autoregressive processes.However,as far as the existing literature is concerned,the moderate deviation is still untestable,as the limiting distributions of the autoregressive coefficient estimators depend on some unknown nuisance parameters.To solve this problem,my thesis innovatively proposes a feasible moderate deviation test,and provides the properties of the test from the perspectives of asymptotic theory and Monte Carlo simulation.Note that the autocorrelation in the random errors is very common.A high-order autoregressive process can be rewritten as a first-order autoregressive process with autocorrelated innovations,under some primitive assumptions.This thesis originally defines an augmented moderate deviation from a unit root and transforms this process into a first-order moderate deviation by defining a kind of quasi difference operators.We derive the convergence rates and liming distributions of the main sample statistics.The results show that the asymptotic behavior of the augmented moderate deviation largely depends on the long-run variance of the error term.To develop the test for the process of augmented moderate deviation,we establish the Wald test,LM test,and LR test.The results show that the Wald,LM and LR statistics all follow a standard Chi-square distribution,which is not related with the functional of the Wiener process and is not related with the long-run variance parameter.This means we do not need to know or estimate the long-run variance in advance.The Monte Carlo simulations validate the theoretical results from a numerical perspective.We apply the new-proposed augmented moderate deviation theory to study the irrational bubble in the real estate market in China's first-and second-tier cities.The results show that after the financial crisis in 2008,the real estate market in China's first-and second-tier cities experienced three rounds of bubbles.The first round is from the beginning of 2010 to the second half of 2010,promoted by the bottoming out of China's economy after the government introduced a strong fiscal policy for the financial crisis.The second round is from the beginning of 2013 to the end of the same year,promoted by the rigid demand of housing and the failure of market expectations.The third round is from the beginning of 2016 to the second half of 2017,which reflects the structural market caused by the slowdown of the economy under the new normal.Furthermore,we can find that the real estate bubble in the first-tier cities is ahead of the corresponding bubble in the secondtier cities,and the former has a higher deviation from the economic fundamental than the latter.It is not hard to understand that the above-mentioned moderate deviation process is essentially driven by the stochastic trend.According to the economic and econometric literature,this stochastic trend measures the potential of economic growth,but ignores the impact of technological advances and economic structural upgrading.Therefore,it is not enough to reflect the inherent logic of real economic growth.Based on this background,we generalize the moderate deviation process to allow for a drift,and then the moderate deviation process is driven by two trends: the stochastic trend and the nonlinear deterministic trend.Both trends can render the process explosive.When the error term has autocorrelation,we employ the simple average of the first few periodograms to estimate the long-run variance and construct the heteroskedasticity and autocorrelation robust standard error.Under the leading-edge asymptotics where the number of periodograms used in the long-run variance estimation is held fixed,we show that the moderate deviation test achieves double robustness: it is asymptotically valid no matter whether the errors are autocorrelated or not,and whether the drift is large,small,or simple not present.We also give some discussions about the long memory innovations and the selection criteria for optimal smoothing parameters.Monte Carlo simulations show that the proposed test has satisfactory size and power performances in finite samples.Applying the test to ten major stock indexed in the pre-2008 financial exuberance period,we find that most indexes are only slightly explosive or not explosive at all,which implies that the bout of the irrational rise was not as serious as previously thought.This is consistent with the perception of Greenspan(2008)that the financial bubble was not so large.Furthermore,this thesis develops the moderate deviation from a unit root to allow for structural breaks in the nonlinear deterministic trend and the stochastic trend.We derive the asymptotic behavior of the main sample statistics under different scenarios of structural breaks,including the structural breaks in the drift term,the structural breaks in the autoregressive coefficient,and the structural breaks in the variance of the random errors.We also introduce how to find the consistent estimators of endogenous structural breaks.In particular,we emphasize the scenario where the autoregressive coefficient structurally changes from a moderate deviation to a unit root.This situation is often regarded as the collapse of the bubble exuberance(see for example,Phillips et al.,2011,Harvey et al.,2016).The results show that the tests based on the autoregressive coefficient have asymptotic robustness with respect to the structural breaks.A series of Monte Carlo simulations lend some support to the conclusion.Finally,we apply the new-proposed tests to investigate the moderate explosiveness in the Bitcoin markets.The results show that the Bitcoin Market Cap in the last quarter of 2017 experienced moderately-explosive exuberance,with a structural break found in November 9.However,although the Bitcoin Market Cap has undergone structural adjustments,it has not changed the irrational growth characteristics in which the autoregressive root has a moderate or slight deviation from the unity.